A337151 a(n) = (n!)^2 * Sum_{k=0..n} (-1)^(n-k) * (k+1) / ((n-k)!)^2.
1, 1, 5, 53, 977, 27649, 1111429, 60147205, 4213400897, 370767834593, 40025019652901, 5199763957426741, 800136077306754385, 143904538461745813153, 29906871652295426507237, 7111902097369951568209349, 1918658066681198636106335489, 582817397769914314847061436225
Offset: 0
Keywords
Programs
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Maple
a:= n-> n!^2 * add((-1)^k*(n-k+1)/k!^2, k=0..n): seq(a(n), n=0..20); # Alois P. Heinz, Jan 27 2021
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Mathematica
Table[n!^2 Sum[(-1)^(n - k) (k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 17}] nmax = 17; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2
Formula
Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - x)^2.
a(n) ~ BesselJ(0,2) * n!^2 * n. - Vaclav Kotesovec, Jul 11 2025