A119429 Expansion of Sum_{k>=0} x^(2*k)/Product_{j=1..k} (1 - j*2*x).
1, 0, 1, 2, 5, 14, 45, 164, 661, 2884, 13461, 66894, 353217, 1977146, 11691481, 72734088, 474172777, 3229062120, 22914397417, 169128976922, 1296276808253, 10300677006854, 84731125615749, 720392483485868, 6321631421441149
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..588
Programs
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PARI
a(n) = sum(k=0, n\2, 2^(n-2*k)*stirling(n-k, k, 2)); \\ Seiichi Manyama, Apr 08 2022
Formula
a(n) = Sum_{k=0..n} S2(k,n-k)*2^(2k-n) where S2(n,k)=A048993(n,k).
a(n) = Sum_{k=0..floor(n/2)} S2(n-k,k)*2^(n-2k) where S2(n,k)=A048993(n,k).
Let E(x) = Sum_{k>=0} x^(2*k)/Product_{j=1..k} (1 - j*2*x), then E(x) = 1 + x^2/(U(0)-x^2), where U(k) = (x-1)^2 - 2*k*x + x^2*(2*k*x + 2*x - 1)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2012
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - x/(1-x*(2*k+2))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013