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

0,2

Links

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

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

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 *)

Prog

(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
(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

Crossrefs

First differences of A001515 and A144301.

Sequence in context: A363908 A167892 A345190 * A201368 A072213 A257741

Adjacent sequences: A144495 A144496 A144497 * A144499 A144500 A144501

Keyword

nonn

Author

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

Status

approved