A321388 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-2)).
1, 1, 1, 4, 19, 144, 1443, 18295, 280918, 5069651, 105147307, 2464296222, 64402891501, 1856989724951, 58560557062508, 2004999890781363, 74069439021212783, 2936703201134924845, 124383305232306494864, 5605027085651919547476, 267759074907470856179460, 13516676464234372267564939
Offset: 0
Keywords
Links
- N. J. A. Sloane, Transforms
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 + x^k)^(k^(k-2)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 -
Maple
a:=series(mul((1+x^k)^(k^(k-2)),k=1..100),x=0,22): seq(coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019
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Mathematica
nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 2)), {k, 1, nmax}], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}] -
PARI
m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-2)))) \\ G. C. Greubel, Nov 09 2018
Formula
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d-1) ) * x^k/k).
a(n) ~ n^(n-2) * (1 + exp(-1)/n + (5*exp(-1)/2 + exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018
Comments