A355723 -id:A355723

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A355721

Square table, read by antidiagonals: the g.f. for row n is given recursively by (2*n-1)*x*R(n,x) = 1 + (2*n-3)*x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112934(k+1)*x^k.

+10
9

1, 1, 2, 1, 2, 6, 1, 2, 10, 26, 1, 2, 14, 74, 158, 1, 2, 18, 138, 706, 1282, 1, 2, 22, 218, 1686, 8162, 13158, 1, 2, 26, 314, 3194, 24162, 110410, 163354, 1, 2, 30, 426, 5326, 53890, 394254, 1708394, 2374078, 1, 2, 34, 554, 8178, 102722, 1019250, 7191018, 29752066, 39456386

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare with A111528, which has a similar definition.

LINKS

Table of n, a(n) for n=0..54.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

FORMULA

Let d(n) = Product_{k = 1..n} 2*k-1 = A001147(n) denote the double factorial of odd numbers.

O.g.f. for row n: R(n,x) = ( Sum_{k >= 0} d(n+k)/d(n)*x^k )/( Sum_{k >= 0} d(n-1+k)/d(n-1)*x^k ).

R(n,x)/(1 - (2*n-1)*x*R(n,x)) = Sum_{k >= 0} d(n+k)/d(n)*x^k.

R(n,x) = 1/(1 + (2*n-1)*x - (2*n+1)*x/(1 + (2*n+1)*x - (2*n+3)*x/(1 + (2*n+3)*x - (2*n+5)*x/(1 + (2*n+5)*x - ... )))).

R(n,x) satisfies the Riccati differential equation 2*x^2*d/dx(R(n,x)) + (2*n-1)*x*R(n,x)^2 - (1 + (2*n-3)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.

Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - (2*n+1)*x/(1 - 4*x/(1 - (2*n+3)*x/(1 - 6*x/(1 - (2*n+5)*x/(1 - ... - 2*m*x/(1 - (2*n+2*m-1)*x/(1 - ... ))))))))), a continued fraction of Stieltjes type.

Row 0: A112934(n+1); Row 1; A000698(n+1).

EXAMPLE

Square array begins

1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...

1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, ...

1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, ...

1, 2, 18, 218, 3194, 53890, 1019250, 21256090, 483426010, 11895873410, ...

1, 2, 22, 314, 5326, 102722, 2197558, 51355514, 1297759918, 35208930050, ...

1, 2, 26, 426, 8178, 176802, 4206618, 108577674, 3011332338, 89141101506, ...

1, 2, 30, 554, 11846, 283042, 7396830, 208569034, 6288011206, 201404591042, ...

...

MAPLE

T := (n, k) -> coeff(series(hypergeom([n+1/2, 1], [], 2*x)/ hypergeom([n-1/2, 1], [], 2*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);

CROSSREFS

Cf. A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355724 (row 4), A355725 (row 5). Cf. A001147, A111528.

KEYWORD

nonn,tabl,easy

AUTHOR

Peter Bala, Jul 15 2022

STATUS

approved

A355722

Row 2 of table A355721.

+10
4

1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, 76246683534, 1968284351178, 54576250392726, 1618348891438242, 51122453577462414, 1714406473587300138, 60843580566100937046, 2278637898592632599682, 89818339421620249242894, 3717488491001699691500298

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

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);

CROSSREFS

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355723 (row 3), A355724 (row 4), A355725 (row 5).

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 15 2022

STATUS

approved

A355724

Row 4 of table A355721.

+10
4

1, 2, 22, 314, 5326, 102722, 2197558, 51355514, 1297759918, 35208930050, 1020115715542, 31432396066106, 1026506419425550, 35428218801977666, 1288967076156307702, 49323199246104202874, 1980947315202528449518, 83342865788161594337282, 3666525676611059535630742

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..18.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

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);

CROSSREFS

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355725 (row 5).

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 15 2022

STATUS

approved

A355725

Row 5 of table A355721.

+10
4

1, 2, 26, 426, 8178, 176802, 4206618, 108577674, 3011332338, 89141101506, 2802596567706, 93232011912426, 3271729161905010, 120810104634555234, 4683805718871051162, 190294015841923438026, 8087576641287426829170, 358981130096398432055682, 16615841072836741527510810

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..18.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

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);

CROSSREFS

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355724 (row 4).

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 15 2022

STATUS

approved

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