A347915 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^exp(x).
1, 1, 4, 24, 150, 1235, 11725, 126987, 1512084, 20313897, 296921623, 4700713787, 80221988726, 1468879687145, 28661345212981, 594457831566757, 13027193829914920, 301079987772726257, 7318797530268562203, 186496088631167771143, 4971371842655844396298, 138384071439982000722737
Offset: 0
Keywords
Programs
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Mathematica
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *) -
PARI
N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(x)))) -
PARI
N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k)))) -
PARI
N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, x^k/(k*(1-x^(2*k))))))) -
PARI
a354507(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/(k*(n-k)!)); a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354507(j)*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 16 2022
Formula
E.g.f.: exp( exp(x) * Sum_{k>=1} A000593(k)*x^k/k ).
E.g.f.: exp( exp(x) * Sum_{k>=1} x^k/(k*(1 - x^(2*k))) ).
a(0) = 1; a(n) = Sum_{k=1..n} A354507(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Aug 16 2022