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

Author

Paul Barry, May 13 2005

Keywords

• easy
• nonn

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

• 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