A064570 Binomial transform of (2n)!.

1, 3, 29, 799, 43353, 3837851, 501393973, 90608944119, 21633834338609, 6593857931708083, 2497877833687172301, 1151118261673522046543, 634098400947597342716809, 411445662820653995008883019

Offset

0,2

Comments

Compare with A229464. - Peter Bala, Sep 25 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

In Maple notation: a(n)=hypergeom([1, 1/2, -n], [], -4), n=0, 1, ...

a(n) = Integral_{x>=0} ((x^4-1)/(x^2-1))^n*exp(-x) dx. - Gerald McGarvey, Oct 14 2006

From Peter Bala, Sep 25 2013: (Start)

a(n) = Sum_{k = 0..n} binomial(n,k)*(2*k)!.

Clearly a(n) is always odd; indeed, a(n) = 1 + 2*n*A229464(n-1) for n >= 1.

Recurrence equation: a(n) = 1 + 2*n*(2*n - 1)*a(n-1) - 2*n*(2*n - 2)*a(n-2) with a(0) = 1 and a(1) = 3.

O.g.f. Sum_{k >= 0} (2*k)!*x^k/(1 - x)^(k + 1) = 1 + 3*x + 29*x^2 + 799*x^3 + .... (End)

Recurrence (homogeneous): a(n) = (4*n^2 - 2*n + 1)*a(n-1) - 2*(n-1)*(4*n-3)*a(n-2) + 4*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Sep 26 2013

a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Sep 26 2013

From Peter Bala, Nov 26 2017: (Start)

E.g.f.: exp(x)*Sum_{n >= 0} A001813(n)*x^n.

a(k) = a(0) (mod k) for all k (by the inhomogeneous recurrence equation).

More generally a(n+k) = a(n) (mod k) for all n and k by an induction argument on n.

It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 9, 3, 1, 3, 9, 9, 3, ... with exact period 5. (End)

Mathematica

Table[Sum[Binomial[n, k]*(2*k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 26 2013 *)

Crossrefs

Cf. A010050, A229464, A001813, A323280.

Sequence in context: A326337 A331389 A243435 * A117264 A256043 A065072

Adjacent sequences: A064567 A064568 A064569 * A064571 A064572 A064573

Keyword

nonn,easy

Author

Karol A. Penson, Sep 20 2001

Status

approved