A132911 a(n) = (n+1)*(2*n)!/2^n.

1, 2, 18, 360, 12600, 680400, 52390800, 5448643200, 735566832000, 125046361440000, 26134689540960000, 6585941764321920000, 1969196587532254080000, 689218805636288928000000, 279133616282697015840000000, 129517997955171415349760000000, 68255984922375335889323520000000

Offset

0,2

Comments

Define T(n,k) = ((1+(-1)^n)/2)*C(k-1+n/2, n/2)*n!/2^(n/2). Then T(n,k) has e.g.f. 1/Sum_{j=0..k} C(k,j)*(-1)^j*x^(2j)/2^j. T(n,1) is A000680 with interpolated zeros. T(n,3) is A132912.

Links

Table of n, a(n) for n=0..16.

Daniele Marchei, Emanuela Merelli, and Andrew Francis, Factorizing the Brauer monoid in polynomial time, arXiv:2402.07874 [math.RA], 2024. See p. 24.

Formula

E.g.f.: 1/(1-x^2+x^4/4) (with interpolated zeros);

a(n)-(n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 05 2012

From Amiram Eldar, Jan 04 2026: (Start)

Sum_{n>=0} 1/a(n) = 2^(1/4)*sqrt(Pi)*BesselI(-3/2, sqrt(2)) + 1.

Sum_{n>=0} (-1)^n/a(n) = sqrt(2)*sin(sqrt(2)) + cos(sqrt(2)) - 1. (End)

Mathematica

Table[(n+1) (2n)!/2^n, {n, 0, 20}] (* Harvey P. Dale, Jun 02 2020 *)

Crossrefs

Cf. A000680, A132912.

Sequence in context: A123311 A349881 A181536 * A291902 A336217 A226837

Adjacent sequences: A132908 A132909 A132910 * A132912 A132913 A132914

Keyword

easy,nonn

Author

Paul Barry, Sep 04 2007

Status

approved