A381346 Expansion of e.g.f. 1/( 1 - sinh(sqrt(2)*x) / sqrt(2) ).
1, 1, 2, 8, 40, 244, 1808, 15632, 154240, 1712656, 21132032, 286800128, 4246266880, 68108302144, 1176458774528, 21772909267712, 429818456473600, 9015349812633856, 200218257664704512, 4693597812326094848, 115820240623410872320, 3000905720793597113344
Offset: 0
Keywords
Programs
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PARI
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j)); a(n) = sum(k=0, n, k!*2^((n-k)/2)*a136630(n, k));
Formula
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^((n-k)/2) * A136630(n,k).
a(n) ~ sqrt(Pi/3) * 2^(n/2 + 1) * n^(n + 1/2) / (arcsinh(sqrt(2))^(n+1) * exp(n)). - Vaclav Kotesovec, Apr 19 2025