A354944 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-n)^(n-k).
1, 0, -10, 60, 1560, -39880, -491760, 45672060, -155935360, -77656158000, 2116774828800, 166585352850620, -11925674437248000, -330617542587341880, 69148933431781898240, -543549949643024194500, -434534462104188331130880, 21521903478880966780355360
Offset: 0
Keywords
Programs
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Mathematica
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! (-n)^(n - k), {k, 0, n}], {n, 0, 17}] Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[(-n)^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}] -
PARI
a(n) = sum(k=0, n, binomial(n,k)^3 * k! * (-n)^(n-k)); \\ Michel Marcus, Jun 12 2022
Formula
a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} (-n)^k * x^k / k!^3.