A350267 a(n) = n*hypergeom([1, 1 - n, -n], [2], 1).
0, 1, 4, 18, 100, 675, 5376, 49294, 510728, 5894109, 74915740, 1039180186, 15613569324, 252501251743, 4371586879128, 80652138666870, 1579212732426256, 32701859350855769, 713914404925713588, 16384896394304282722, 394340620941231415540, 9929838681717090607611
Offset: 0
Keywords
Programs
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Maple
A350267 := n -> n*hypergeom([1, 1 - n, -n], [2], 1): seq(simplify(A350267(n)), n = 0..21); # Or: egf := (exp(x/(1-x)) - exp(x))/x: ser := series(egf, x, 23): seq(n!*coeff(ser, x, n), n = 0..21); # Peter Luschny, Jul 01 2025
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Mathematica
a[n_] := Sum[Binomial[n, k]^2 * k!/(n - k + 1), {k, 1, n}]; Array[a, 22, 0] (* Amiram Eldar, Jan 09 2022 *) -
PARI
a(n) = sum(k=1, n, binomial(n, k)^2 * k! / (n - k + 1)); \\ Michel Marcus, Jan 09 2022
Formula
a(n) = n*A247499(n - 1) for n >= 1.
a(n) = Sum_{k=1..n} binomial(n, k)^2 * k! / (n - k + 1).
E.g.f.: (exp(x/(1-x)) - exp(x))/x. - Vladimir Kruchinin, Seiichi Manyama, Jul 01 2025
Extensions
Definition changed to a(0) = 0 by Peter Luschny, Jul 01 2025