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A363308 Expansion of g.f. C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%I A363308 #19 Nov 22 2024 06:48:01
%S A363308 1,1,5,26,141,790,4542,26668,159333,966038,5930678,36801660,230491410,
%T A363308 1455283172,9253674120,59209786992,380961295445,2463303690790,
%U A363308 15998687418030,104325569140156,682768883525830,4483232450501492,29527005540912660,195006621974036808
%N A363308 Expansion of g.f. C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%C A363308 Compare the g.f. A(x) = C(x*C(x)^3) to the identity C(-x*C(x)^3) = 1/C(x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%C A363308 Conjecture: a(n) is odd iff n is a power of 2 or n = 0.
%H A363308 Paul D. Hanna, <a href="/A363308/b363308.txt">Table of n, a(n) for n = 0..400</a>
%F A363308 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, C(x) is the g.f. of the Catalan numbers (A000108).
%F A363308 (1) A(x) = C(x*C(x)^3), where C(x) = (1 - sqrt(1-4*x))/(2*x).
%F A363308 (2) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = C( x*B(x) * C(x*B(x))^3 ) is the g.f. of A033296.
%F A363308 (3) a(n) = Sum_{k=1..n} 3*k* binomial(2*k+1,k) * binomial(2*n+k,n-k) / ((2*k+1)*(2*n+k)) for n > 0, with a(0) = 1.
%F A363308 D-finite with recurrence 4*n*(n-1)*(21687905*n +56141583)*(n+1)*a(n) +2*n*(n-1) *(43375810*n^2 -6022811713*n +10976463649)*a(n-1) -(n-1) *(15429963345*n^3 -200018809315*n^2 +658353214412*n -632905646028)*a(n-2) +(102558230760*n^4 -1409936457473*n^3 +6909548744112*n^2 -14414518702669*n +10812683474490)*a(n-3) +(-212869593020*n^4 +3377685007909*n^3 -20069314453381*n^2 +52902205420466*n -52146873039204)*a(n-4) +4*(2*n-11) *(10773532140*n^3 -171459615587*n^2 +902576783797*n -1572525214995)*a(n-5) +4*(n-6) *(208500820*n -955419151)*(2*n-11) *(2*n-13)*a(n-6)=0. - _R. J. Mathar_, Nov 22 2024
%e A363308 G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 141*x^4 + 790*x^5 + 4542*x^6 + 26668*x^7 + 159333*x^8 + 966038*x^9 + 5930678*x^10 + ...
%e A363308 such that A(x) = C(x*C(x)^3), where
%e A363308 C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... + A000108(n)*x^n + ...
%e A363308 x*C(x)^3 = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + ... + A000245(n)*x^n + ...
%e A363308 Note that x*C(x)^3 = (C(x) - 1)*(1-x)/x - 1.
%e A363308 Also, the g.f. of related sequence A033296 begins
%e A363308 B(x) = 1 + x + 6*x^2 + 42*x^3 + 326*x^4 + 2706*x^5 + 23526*x^6 + ...
%e A363308 where A(x) = B(x/A(x)), B(x) = A(x*B(x)) = C(x*B(x)*C(x*B(x))^3).
%o A363308 (PARI) {a(n) = if(n==0,1, sum(k=1,n, 3*k* binomial(2*k+1,k) * binomial(2*n+k,n-k) / ((2*k+1)*(2*n+k)) ) )}
%o A363308 for(n=0, 30, print1(a(n), ", "))
%o A363308 (PARI) /* G.f. A(x) = C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 */
%o A363308 {a(n) = my(C = (1 - sqrt(1 - 4*x +x^2*O(x^n)))/(2*x)); polcoeff( subst(C, x, x*C^3), n)}
%o A363308 for(n=0, 30, print1(a(n), ", "))
%Y A363308 Cf. A127632, A153294, A033296, A000108 (C(x)), A000245 (x*C(x)^3).
%Y A363308 Cf. A363309, A363111.
%K A363308 nonn
%O A363308 0,3
%A A363308 _Paul D. Hanna_, May 28 2023