A111140 - OEIS

%I #14 Sep 08 2022 08:45:20

%S 1,1,3,13,71,466,3582,31641,316171,3526606,43421978,584716386,

%T 8544649478,134622445348,2274031087772,40987164702945,784981384215795,

%U 15916200367695510,340548893413909410,7666975604019750630

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

%H G. C. Greubel, <a href="/A111140/b111140.txt">Table of n, a(n) for n = 0..250</a>

%F 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

%F 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

%t 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 *)

%o (PARI) {a(n) = (n!/(n+1))*sum(k=0,n, binomial(n+k-1,k)/k!)};

%o vector(20, n, n--; a(n)) \\ _G. C. Greubel_, Feb 07 2019

%o (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

%o (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

%o (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

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Oct 17 2005