A107232 Expansion of (1+x*c(x^2))^3/sqrt(1-4*x^2), c(x) the g.f. of A000108.

1, 3, 5, 10, 18, 35, 65, 126, 238, 462, 882, 1716, 3300, 6435, 12441, 24310, 47190, 92378, 179894, 352716, 688636, 1352078, 2645370, 5200300, 10192588, 20058300, 39373700, 77558760, 152443080, 300540195, 591385545, 1166803110, 2298248550

Offset

0,2

Comments

An inverse Chebyshev transform of C(3,n)=(1,3,3,1,0,0,0,...), where g(x)->(1/sqrt(1-4x^2))g(xc(x^2)). In general, (1+xc(x^2))^r/sqrt(1-4x^2) has general term a(n)=sum{k=0..floor(n/2), binomial(n,k)*binomial(r,n-2k)}, r>0.

Links

Table of n, a(n) for n=0..32.

Piera Manara and Claudio Perelli Cippo, The fine structure of 4321 avoiding involutions and 321 avoiding involutions, PU. M. A. Vol. 22 (2011), 227-238. - From N. J. A. Sloane, Oct 13 2012

Formula

a(n)=sum{k=0..floor(n/2), binomial(n, k)*binomial(3, n-2k)}.

D-finite with recurrence: -(n+3)*(3*n-2)*a(n) +12*n*a(n-1) +4*(3*n+1)*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 04 2017

a(n) ~ 2^(n + 5/2) / sqrt(Pi*n). - Vaclav Kotesovec, Sep 28 2020

Crossrefs

Sequence in context: A094986 A154949 A318248 * A134522 A001445 A192860

Adjacent sequences: A107229 A107230 A107231 * A107233 A107234 A107235

Keyword

easy,nonn

Author

Paul Barry, May 13 2005

Status

approved