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a(n) = Sum_{k=floor((n-1)/4)..(n-1)} binomial(2*k,n-2*k-1)*C(k)}, , where C(k) are the Catalan numbers (A000108).
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The numbers 12, 56, 992, 16256, 67100672,… ... ( A139256(n)), , twice even perfect numbers, ) are in the sequence because they are oblong (A139256(n) = 2^k*(2^k-1) with 2^k-1 Mersenne prime) and sigma(A139256(n)) = sigma(2^k*(2^k-1)) = sigma(2^k )*sigma( (2^k-1) = (2^(k+1)-1)*2^(k+1)/2, is a triangular number.
2 is in the sequence because 2=1*2 is oblong, and sigma(2) = 1+2 = 3 = 2*3/2 is triangular.
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G.f.: x*7*(43+17*x)/(1-x)^2. (Corrected by Vincenzo Librandi, Apr 11 2015)
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a(n) = (Sum_{k=0..(n+1)} binomial(2*k-2,k)*2^(n-k+1)*binomial(2*n-k,n-k+1))/n, a(0)=1.
(PARI) my(x='x+O('x^50)); Vec(2-x*2/(1-(1-8*x)^(1/4))) \\ G. C. Greubel, Jun 03 2017
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(PARI) my(x='x+O('x^50)); Vec((2*x+1)/(2*sqrt(4*x^2-8*x+1)) + 1/2) \\ G. C. Greubel, Jun 03 2017
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