A111140 a(n) = (n!/(n+1))*Sum_{k=0..n} binomial(n+k-1,k)/k!.

1, 1, 3, 13, 71, 466, 3582, 31641, 316171, 3526606, 43421978, 584716386, 8544649478, 134622445348, 2274031087772, 40987164702945, 784981384215795, 15916200367695510, 340548893413909410, 7666975604019750630

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

Formula

Recurrence: n*(n+1)*(n^3 - 12*n^2 + 37*n - 32)*a(n) = 2*n*(n^5 - 10*n^4 + 14*n^3 + 45*n^2 - 92*n + 30)*a(n-1) - (n-1)*(n^6 - 6*n^5 - 39*n^4 + 294*n^3 - 612*n^2 + 488*n - 120)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*(2*n - 5)*(n^3 - 9*n^2 + 16*n - 6)*a(n-3). - Vaclav Kotesovec, Nov 27 2017

a(n) ~ exp(2*sqrt(n) - n + 1/2) * n^(n - 3/4) / sqrt(2) * (1 - 17/(48*sqrt(n))). - Vaclav Kotesovec, Nov 27 2017

Mathematica

f[n_]:= n!/(n+1)*Sum[Binomial[n+k-1, k]/k!, {k, 0, n}]; Table[f[n], {n, 0, 10}] (* Robert G. Wilson v, Oct 21 2005 *)

Prog

(PARI) {a(n) = (n!/(n+1))*sum(k=0, n, binomial(n+k-1, k)/k!)};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019
(Magma) [(Factorial(n)/(n+1))*(&+[Binomial(n+k-1, k)/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Feb 07 2019
(Sage) [(factorial(n)/(n+1))*sum(binomial(n+k-1, k)/factorial(k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Feb 07 2019
(GAP) List([0..20], n-> (Factorial(n)/(n+1))*Sum([0..n], k-> Binomial(n+k-1, k)/Factorial(k)) ) # G. C. Greubel, Feb 07 2019

Crossrefs

Sequence in context: A192239 A192936 A000261 * A302699 A137983 A327677

Adjacent sequences: A111137 A111138 A111139 * A111141 A111142 A111143

Keyword

easy,nonn

Author

Vladeta Jovovic, Oct 17 2005

Status

approved