A144498 Column 2 of array in A144502.

1, 5, 30, 229, 2165, 24576, 326515, 4976315, 85630914, 1642623355, 34762642871, 804650577600, 20224019536825, 548535471681029, 15969883030969470, 496754110707779461, 16441934503725675485, 576991048929859964160, 21399021201104749243099, 836326710446071005267035

Offset: 0

Author

David Applegate and N. J. A. Sloane, Dec 13 2008

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

First differences of A001515 and A144301.

Programs

Magma

A144498:= func< n | (&+[Binomial(n,k)*Factorial(n+k+1)/(2^k*Factorial(n)): k in [0..n]]) >;
[A144498(n): n in [0..30]]; // G. C. Greubel, Oct 07 2023

Maple

f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..60)];

Mathematica

Table[Sum[(n+k+1)!/((n-k)!*k!*2^k), {k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2013 *)

SageMath

def A144498(n): return sum(binomial(n,k)*rising_factorial(n+1,k+1)//2^k for k in range(n+1))
[A144498(n) for n in range(31)] # G. C. Greubel, Oct 07 2023

Formula

a(n) = A001515(n+1) - A001515(n).

a(n) = A144301(n+2) - A144301(n+1).

E.g.f.: (1 - 2*x + 2*x*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(1-2*x)^2. - Sergei N. Gladkovskii, Oct 06 2012

G.f.: (1-x)/(x*Q(0)) - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013

a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n-1). - Vaclav Kotesovec, Oct 08 2013

G.f.: T(0)/x- 1/x, where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2013

(2*n-1)*a(n) = (4*n^2 + 1)*a(n-1) + (2*n+1)*a(n-2). - G. C. Greubel, Oct 07 2023