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%I #27 Mar 18 2024 05:24:28
%S 1,1,2,6,19,64,227,832,3126,11980,46646,184003,733783,2953434,
%T 11982265,48949631,201182110,831292029,3451336467,14390479996,
%U 60232976244,252992172572,1066000599632,4504710385216,19086728370308,81069926894797
%N G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x)^3).
%H G. C. Greubel, <a href="/A181734/b181734.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..[n/2]} C(n+k, k)*C(n+k+1, n-2k)/(n+1).
%F G.f.: A(x) = (1/x)*Series_Reversion( x/(1+x) - x^3 ).
%F G.f. satisfies:
%F (1) A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 + x)/(1 - x^2 - x^3).
%F (2) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*x^k*A(x)^(2k)] * x^n/n ).
%F (3) A(x) = exp( Sum_{n>=1} [Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^(2k)]*(1-x*A(x)^2)^(2*n+1)* x^n/n ).
%F Recurrence: 23*(n-1)*n*(n+1)*(429*n^2 - 1903*n + 2004)*a(n) = 3*(n-1)*n*(12870*n^3 - 63525*n^2 + 87653*n - 28312)*a(n-1) + 3*(n-1)*(3861*n^4 - 24849*n^3 + 59286*n^2 - 61456*n + 22956)*a(n-2) + 3*(23166*n^5 - 207009*n^4 + 712578*n^3 - 1173947*n^2 + 916800*n - 266772)*a(n-3) - 3*(n-3)*(3*n - 10)*(3*n - 5)*(429*n^2 - 1045*n + 530)*a(n-4). - _Vaclav Kotesovec_, Sep 18 2013
%F a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 1/46*(45 + 36*sqrt(3) + 23*sqrt(3429/529 + (3792*sqrt(3))/529)) = 4.50735893936524052... is the root of the equation 27 - 162*d - 27*d^2 - 90*d^3 + 23*d^4 = 0 and c = 1/(2*sqrt(-29 - 17*sqrt(3) + 5*sqrt(69 + 40*sqrt(3)))) = 0.842957337580479516110291505734... - _Vaclav Kotesovec_, Sep 18 2013, updated Mar 18 2024
%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 227*x^6 +...
%e The logarithm of the g.f. may be expressed as the series:
%e log(A(x)) = (1 + x*A(x)^2)*x +
%e (1 + 4*x*A(x)^2 + x^2*A(x)^4)*x^2/2 +
%e (1 + 9*x*A(x)^2 + 9*x^2*A(x)^4 + x^3*A(x)^6)*x^3/3 +
%e (1 + 16*x*A(x)^2 + 36*x^2*A(x)^4 + 16*x^3*A(x)^6 + x^4*A(x)^8)*x^4/4 +...
%e which involves the squared binomial coefficients.
%t Table[Sum[Binomial[n+k,k]*Binomial[n+k+1, n-2k]/(n+1),{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 18 2013 *)
%o (PARI) {a(n)=sum(k=0, n\2, binomial(n+k, k)*binomial(n+k+1, n-2*k))/(n+1)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=(1+x*A)*(1+x^2*A^3));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(x*A^2+x*O(x^n))^j)*x^m/m)));polcoeff(A, n, x)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(j=0, n, binomial(m+j, j)^2*(x*A^2+x*O(x^n))^j)*(1-x*A^2)^(2*m+1)*x^m/m)));polcoeff(A, n, x)}
%Y Cf. A036765.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 14 2011
%E Name changed slightly by _Paul D. Hanna_, Nov 14 2012
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