| Displaying 1-9 of 9 results found.
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1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, 76246683534, 1968284351178, 54576250392726, 1618348891438242, 51122453577462414, 1714406473587300138, 60843580566100937046, 2278637898592632599682, 89818339421620249242894, 3717488491001699691500298
FORMULA
O.g.f: A(x) = ( Sum_{k >= 0} d(k+2)/d(2)*x^k )/( Sum_{k >= 0} d(k+1)/d(1)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n). A(x)= 1/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 3*x*A(x)^2 - (1 + x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - 9*x/(1 - ... - 2*n*x/(1 - (2*n+3)*x )))))))), a continued fraction of Stieltjes type.
MAPLE
n := 2: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);
1, 2, 18, 218, 3194, 53890, 1019250, 21256090, 483426010, 11895873410, 314834663250, 8918883839450, 269367643864250, 8643467766472450, 293770652998691250, 10546424484691428250, 398914704362503668250, 15860639479547463637250, 661439858772303085871250, 28874834455755565593004250
FORMULA
O.g.f: A(x) = ( Sum_{k >= 0} d(k+3)/d(3)*x^k )/( Sum_{k >= 0} d(k+2)/d(2)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n). A(x) = 1/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 5*x*A(x)^2 - (1 + 3*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - 11*x/(1 - ... - 2*n*x/(1 - (2*n+5)*x )))))))), a continued fraction of Stieltjes type.
MAPLE
n := 3: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);
1, 2, 22, 314, 5326, 102722, 2197558, 51355514, 1297759918, 35208930050, 1020115715542, 31432396066106, 1026506419425550, 35428218801977666, 1288967076156307702, 49323199246104202874, 1980947315202528449518, 83342865788161594337282, 3666525676611059535630742
FORMULA
O.g.f: A(x) = ( Sum_{k >= 0} d(k+4)/d(4)*x^k )/( Sum_{k >= 0} d(k+3)/d(3)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n). A(x) = 1/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - 13*x/(1 + 13*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 7*x*A(x)^2 - (1 + 5*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - 13*x/(1 - ... - 2*n*x/(1 - (2*n+7)*x )))))))), a continued fraction of Stieltjes-type.
MAPLE
n := 4: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);
1, 2, 26, 426, 8178, 176802, 4206618, 108577674, 3011332338, 89141101506, 2802596567706, 93232011912426, 3271729161905010, 120810104634555234, 4683805718871051162, 190294015841923438026, 8087576641287426829170, 358981130096398432055682, 16615841072836741527510810
FORMULA
O.g.f: A(x) = ( Sum_{k >= 0} d(k+5)/d(5)*x^k )/( Sum_{k >= 0} d(k+4)/d(4)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n). A(x) = 1/(1 + 9*x - 11*x/(1 + 11*x - 13*x/(1 + 13*x - 15*x/(1 + 15*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 9*x*A(x)^2 - (1 + 7*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - 15*x/(1 - ... - 2*n*x/(1 - (2*n+9)*x )))))))), a continued fraction of Stieltjes-type.
MAPLE
n := 5: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);
Square table, read by antidiagonals: the g.f. for row n is given recursively by (3*n-1)*x*R(n,x) = 1 + (3*n-4)*x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112936(k+1)*x^k.
+10
4
1, 1, 3, 1, 3, 15, 1, 3, 24, 111, 1, 3, 33, 282, 1131, 1, 3, 42, 507, 4236, 14943, 1, 3, 51, 786, 9609, 76548, 243915, 1, 3, 60, 1119, 17736, 212835, 1608864, 4742391, 1, 3, 69, 1506, 29103, 459768, 5350785, 38488152, 106912131, 1, 3, 78, 1947, 44196, 859143, 13333488
COMMENTS
Compare with A111528 and A355721, which have similar definitions and properties.
FORMULA
Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers). O.g.f. for row n >= 0: R(n,x) = ( Sum_{k >= 0} t(n+k)/t(n)*x^k )/( Sum_{k >= 0} t(n-1+k)/t(n-1)*x^k ).
R(n,x)/(1 - (3*n-1)*x*R(n,x)) = Sum_{k >= 0} t(n+k)/t(n)*x^k.
R(n,x) = 1/(1 + (3*n-1)*x - (3*n+2)*x/(1 + (3*n+2)*x - (3*n+5)*x/(1 + (3*n+5)*x - (3*n+8)*x/(1 + (3*n+8)*x - ... )))) (continued fraction).
R(n,x) satisfies the Riccati differential equation 3*x^2*d/dx(R(n,x)) + (3*n-1)*x*R(n,x)^2 - (1 + (3*n-4)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.
Applying Stokes 1982 gives R(n,x) = 1/(1 - 3*x/(1 - (3*n+2)*x/(1 - 6*x/(1 - (3*n+5)*x/(1 - 9*x/(1 - (3*n+8)*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.
EXAMPLE
Square array begins
1, 3, 15, 111, 1131, 14943, 243915, 4742391, 106912131, ...
1, 3, 24, 282, 4236, 76548, 1608864, 38488152, 1032125136, ...
1, 3, 33, 507, 9609, 212835, 5350785, 149961675, 4628365305, ...
1, 3, 42, 786, 17736, 459768, 13333488, 425600976, 14791250688, ...
1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, ...
1, 3, 60, 1506, 44196, 1458588, 52917360, 2080630776, 87823112496, ...
1, 3, 69, 1947, 63501, 2311563, 91949469, 3943276347, 180679742061, ...
1, 3, 78, 2442, 87504, 3477360, 150259200, 6970190160, 344116224960, ...
MAPLE
T := (n, k) -> coeff(series(hypergeom([n+2/3, 1], [], 3*x)/ hypergeom([n-1/3, 1], [], 3*x), x, 21), x, k):
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
1, 3, 24, 282, 4236, 76548, 1608864, 38488152, 1032125136, 30670171248, 1000637672064, 35571839009952, 1368990872569536, 56720594992438848, 2517761078627172864, 119222916630934484352, 5999613754698100628736, 319763269764299852744448, 17994913747767982690289664
FORMULA
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+1)/t(1)*x^k )/( Sum_{k >= 0} t(k)/t(0)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers). A(x)/(1 - 2*x*A(x)) = Sum_{k >= 0} t(k+1)/t(1)*x^k.
A(x) = 1/(1 + 2*x - 5*x/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 2*x*A(x)^2 - (1 - x)*A(x) + 1 = 0 with A(0) = 1.
Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 5*x/(1 - 6*x/(1 - 8*x/(1 - 9*x/(1 - 11*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.
MAPLE
n := 1: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);
1, 3, 33, 507, 9609, 212835, 5350785, 149961675, 4628365305, 155913036915, 5692874399025, 224034935130075, 9456933847187625, 426402330032719875, 20460268520575152225, 1041301103429870128875, 56040353252589013121625, 3180443637298592493577875, 189863589771186976073108625
FORMULA
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+2)/t(2)*x^k )/( Sum_{k >= 0} t(k+1)/t(1)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers). A(x)/(1 - 5*x*A(x)) = Sum_{k >= 0} t(k+2)/t(2)*x^k.
A(x) = 1/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 5*x*A(x)^2 - (1 + 2*x)*A(x) + 1 = 0 with A(0) = 1.
Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 8*x/(1 - 6*x/(1 - 11*x/(1 - 9*x/(1 - 14*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.
MAPLE
n := 2: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);
1, 3, 42, 786, 17736, 459768, 13333488, 425600976, 14791250688, 555381292800, 22398626084352, 965768866650624, 44347055502428160, 2161455366606034944, 111489317304231616512, 6069676735484389779456, 347921629212782938472448, 20950823605616500202323968, 1322561808699778749456678912
FORMULA
Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers). O.g.f.: A(x) = ( Sum_{k >= 0} t(k+3)/t(3)*x^k )/( Sum_{k >= 0} t(k+2)/t(2)*x^k ).
A(x)/(1 - 8*x*A(x)) = Sum_{k >= 0} t(k+3)/t(3)*x^k.
A(x) = 1/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - 17*x/(1 + 17*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 8*x*R(n,x)^2 - (1 + 5*x)*R(n,x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 11*x/(1 - 6*x/(1 - 14*x/(1 - 9*x/(1 - 17*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.
MAPLE
n := 3: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);
1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, 1595167432599, 70315835952471, 3293268346004439, 163337193581191575, 8554718468806548951, 471976737725208306327, 27369722655919760451159, 1664858070989667129693975, 106029602841882346657155543
FORMULA
Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers). O.g.f.: A(x) = ( Sum_{k >= 0} t(k+4)/t(4)*x^k )/( Sum_{k >= 0} t(k+3)/t(3)*x^k ).
A(x)/(1 - 11*x*A(x)) = Sum_{k >= 0} t(k+4)/t(4)*x^k.
A(x) = 1/(1 + 11*x - 14*x/(1 + 14*x -17*x/(1 + 17*x - 20*x/(1 + 20*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 11*x*A(x)^2 - (1 + 8*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 14*x/(1 - 6*x/(1 - 17*x/(1 - 9*x/(1 - 20*x/(1 - 12*x/(1 - 23*x/(1 - ...))))))))), a continued fraction of Stieltjes type.
MAPLE
n := 4: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);
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