A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.
1, 2, 7, 46, 485, 7066, 130987, 2946182, 77923561, 2369742130, 81467904431, 3124302688222, 132237820201357, 6123150708289226, 307903794151741075, 16709463201832993846, 973385368533058021457, 60583668821975488285282, 4012342371757905842648791, 281735471040327667890013070
Offset: 0
Keywords
Programs
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Maple
gf := exp(x)*(1 - sqrt(1 - 4*x))/(2*x): ser := series(gf, x, 24): seq(n!*coeff(ser, x, n), n = 0..19); # Alternative: a := n -> `if`(n < 4, [1, 2, 7, 46][n + 1], ((4*n^2 - 12*n + 8)*a(n - 3) - (8*n^2 - 13*n + 5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1)): seq(a(n), n = 0..19); # Peter Luschny, Nov 23 2021 # Alternative seq(simplify(hypergeom([-n, 1/2, 1], [2], -4)), n = 0..19); # Peter Bala, Mar 13 2025
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Mathematica
Table[Sum[Binomial[n, j]*CatalanNumber[j]*j!, {j, 0, n}], {n, 0, 20}] -
PARI
a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k!); \\ Michel Marcus, Nov 23 2021
Formula
a(n) ~ 2^(2*n + 1/2) * n^(n-1) / exp(n - 1/4).
From Peter Luschny, Nov 23 2021: (Start)
a(n) = n! * [x^n](exp(x)*(1 - sqrt(1 - 4*x))/(2*x)).
a(n) = (4*(n-1)*(n-2)*a(n - 3) - (n-1)*(8*n-5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1) for n >= 4.
a(n-1) = A224500(n) / n for n >= 1. (End)
a(n) = hypergeom([-n, 1/2, 1], [2], -4). - Peter Bala, Mar 13 2025
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