A165433 A transform of the double factorial numbers A001147.

1, 1, 2, 3, 7, 14, 39, 97, 308, 897, 3139, 10304, 38997, 140893, 570002, 2230599, 9567979, 40091222, 181203603, 805962157, 3819522284, 17912075229, 88646095447, 435959031488, 2245454002137, 11530035000169, 61627679281154

Offset

0,3

Comments

Hankel transform is A000178.

Links

G. C. Greubel, Table of n, a(n) for n = 0..800

Formula

G.f.: 1/(1-x-x^2-2x^4/(1-x-5x^2-12x^4/(1-x-9x^2-30x^4/(1-x-13x^2-56x^4/(1-.... (continued fraction);

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(2k)!/(k!*2^k).

Conjecture: 2*a(n) -3*a(n-1) +(3-2*n)*a(n-2) +(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 14 2011

G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013

a(n) ~ 2^(-1/2) * exp(sqrt(n)/2 - n/2 + 1/16) * n^(n/2) * (1 + 121/(192*sqrt(n))). - Vaclav Kotesovec, Apr 18 2024

Maple

a:=proc(n) add(binomial(n-k, k)*factorial(2*k)/(factorial(k)*2^k), k=0..floor(n/2)) end proc: seq(a(n), n=0..30); # Muniru A Asiru, Oct 20 2018

Mathematica

Table[Sum[Binomial[n-k, k]*(2*k)!/(k!*2^k), {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Oct 20 2018 *)

Prog

(PARI) vector(30, n, n--; sum(k=0, floor(n/2), binomial(n-k, k)*(2*k)!/(k!*2^k))) \\ G. C. Greubel, Oct 20 2018
(Magma) [(&+[Binomial(n-k, k)*Factorial(2*k)/(Factorial(k)*2^k): k in [0.. Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 20 2018

Crossrefs

Sequence in context: A006785 A274538 A113182 * A238728 A271718 A222626

Adjacent sequences: A165430 A165431 A165432 * A165434 A165435 A165436

Keyword

easy,nonn

Author

Paul Barry, Sep 18 2009

Status

approved