A346750 Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).
0, 0, 1, 3, 3, -20, -135, -189, 3598, 33300, 39105, -2164085, -23831214, -5268042, 3038813869, 36984819795, -59749871880, -8207734934984, -105142191601887, 482549202944307, 37754304692254030, 489494512692093090, -4466445363328684659, -271973408844483808517
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..457
Programs
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Mathematica
nmax = 23; CoefficientList[Series[Log[1 + x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]! a[0] = 0; a[n_] := a[n] = Binomial[n, 2] - (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Formula
a(0) = 0; a(n) = binomial(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023
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