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A181734
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G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x)^3).
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8
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1, 1, 2, 6, 19, 64, 227, 832, 3126, 11980, 46646, 184003, 733783, 2953434, 11982265, 48949631, 201182110, 831292029, 3451336467, 14390479996, 60232976244, 252992172572, 1066000599632, 4504710385216, 19086728370308, 81069926894797
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..[n/2]} C(n+k, k)*C(n+k+1, n-2k)/(n+1).
G.f.: A(x) = (1/x)*Series_Reversion( x/(1+x) - x^3 ).
G.f. satisfies:
(1) A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 + x)/(1 - x^2 - x^3).
(2) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*x^k*A(x)^(2k)] * x^n/n ).
(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 ).
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
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
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 227*x^6 +...
The logarithm of the g.f. may be expressed as the series:
log(A(x)) = (1 + x*A(x)^2)*x +
(1 + 4*x*A(x)^2 + x^2*A(x)^4)*x^2/2 +
(1 + 9*x*A(x)^2 + 9*x^2*A(x)^4 + x^3*A(x)^6)*x^3/3 +
(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 +...
which involves the squared binomial coefficients.
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n)=sum(k=0, n\2, binomial(n+k, k)*binomial(n+k+1, n-2*k))/(n+1)}
(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)}
for(n=0, 30, print1(a(n), ", "))
(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)}
(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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