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A346312
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / n^2).
1
1, -1, -1, 5, 28, 724, 36, 220716, -1255680, 110979072, 2530310400, 1193835283200, -24457819622400, 21656019855744000, 899271273253248000, 474367063601421849600, 45822442913828595302400, 28365278076547150440038400, 2614371018285307258994688000
OFFSET
0,4
FORMULA
a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} (binomial(n,k) * k!)^2 * ( Sum_{d|k} 1 / (k/d)^(2*d-1) ) * a(n-k).
MATHEMATICA
nmax = 18; CoefficientList[Series[Product[(1 - x^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[(Binomial[n, k] k!)^2 Sum[1/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jul 13 2021
STATUS
approved