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%I A107232 #19 Feb 23 2024 17:12:32
%S A107232 1,3,5,10,18,35,65,126,238,462,882,1716,3300,6435,12441,24310,47190,
%T A107232 92378,179894,352716,688636,1352078,2645370,5200300,10192588,20058300,
%U A107232 39373700,77558760,152443080,300540195,591385545,1166803110,2298248550
%N A107232 Expansion of (1+x*c(x^2))^3/sqrt(1-4*x^2), c(x) the g.f. of A000108.
%C A107232 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.
%H A107232 Piera Manara and Claudio Perelli Cippo, <a href="http://puma.dimai.unifi.it/22_2/manara_perelli-cippo.pdf">The fine structure of 4321 avoiding involutions and 321 avoiding involutions</a>, PU. M. A. Vol. 22 (2011), 227-238. - From _N. J. A. Sloane_, Oct 13 2012
%F A107232 a(n)=sum{k=0..floor(n/2), binomial(n, k)*binomial(3, n-2k)}.
%F A107232 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
%F A107232 a(n) ~ 2^(n + 5/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Sep 28 2020
%K A107232 easy,nonn
%O A107232 0,2
%A A107232 _Paul Barry_, May 13 2005

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