A382847 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (Stirling2(n,k) * k!)^2.
1, 1, 14, 579, 48044, 6647405, 1379024730, 400315753159, 154879704709784, 77018569697097009, 47863427797633958630, 36348262891572161261963, 33119479438137288670256964, 35660343372397246917403353013, 44791475616825872944740798413234, 64911462519379469821754507087299215
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[Binomial[n + k - 1, k] (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 15}] Table[(n!)^2 SeriesCoefficient[1/(Exp[x] + Exp[y] - Exp[x + y])^n, {x, 0, n}, {y, 0, n}], {n, 0, 15}] -
PARI
a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 2))^2); \\ Seiichi Manyama, Apr 06 2025
Formula
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y))^n.
a(n) == 0 (mod n) for n > 0. - Seiichi Manyama, Apr 06 2025
a(n) ~ c * (r*(1+r)*(1 + 2*r + 2*sqrt(r*(1+r))))^n * n^(2*n) / exp(2*n), where r = 0.78386040488712123296193324113250946749673854534386788724235... is the root of the equation r = (1+r) * (1 + 1/(r*LambertW(-exp(-1/r)/r)))^2 and c = 0.947509273452712778524331973956110163137127694168427319... - Vaclav Kotesovec, Apr 08 2025