OFFSET
0,3
COMMENTS
Compare g.f. to a g.f. B(x) of Catalan numbers (A000108):
B(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k) *x^k* B(x)^(2k)].
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..[n/2]} C(n+k, 2k)*C(n+1, n-2k)/(n+1).
G.f. A(x) satisfies:
(1) A(x) = 1/sqrt{ [1-x - x^2*A(x)^2]^2 - 4*x^3*A(x)^2 }.
(2) A(x) = (1/x)*Series_Reversion{ x*(1-x^2)^2/[sqrt((1-x^2)^3 + x^2*(1+x^2)^2) + x*(1+x^2)] }.
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) satisfies: 1-x^2 = (1-x^2)^2*G(x)^2 - 2*x*(1+x^2)*G(x).
(4) A(x) = Sum_{n>=0} x^n*(1 - x*A(x)^2)^(2*n+1) * [Sum_{k>=0} C(n+k,k)^2 *x^k*A(x)^(2k)].
(5) A(x) = Sum_{n>=0} x^(2n)*A(x)^(2n)*[Sum_{k>=0} C(n+k,k)^2*x^k].
(6) A(x) = Sum_{n>=0} x^(2n)*A(x)^(2n)*[Sum_{k=0..n} C(n,k)^2*x^k] /(1-x)^(2n+1).
(7) A(x) = Sum_{n>=0} (2n)!/n!^2 * x^(3n)*A(x)^(2n)/(1-x-x^2*A(x)^2)^(2n+1).
Recurrence: 8*n*(n+1)*(2*n+1)*(5221*n^4 - 34247*n^3 + 79127*n^2 - 74851*n + 23310)*a(n) = 12*n*(26105*n^6 - 171235*n^5 + 398694*n^4 - 384054*n^3 + 116038*n^2 + 18484*n - 9792)*a(n-1) + 6*(31326*n^7 - 252471*n^6 + 759709*n^5 - 1008483*n^4 + 452035*n^3 + 190224*n^2 - 218420*n + 50400)*a(n-2) + (n-2)*(1164283*n^6 - 8801364*n^5 + 23941663*n^4 - 27089928*n^3 + 8937814*n^2 + 3724572*n - 2026080)*a(n-3) - 10*(n-3)*(n-2)*(2*n-7)*(5221*n^4 - 13363*n^3 + 7712*n^2 + 1546*n - 1440)*a(n-4). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ (r^(1/2-n) * sqrt((1 + 7*r^6*s^6 - 2*r^7*s^8 - 3*r^2*(-1+s^2) - 5*r^4*s^2*(-1+s^2) + r*(-3+2*s^2) - r^3*(1 + 4*s^2 + 6*s^4) + r^5*(-9*s^4 + 6*s^6))/(r^2*(1 - r - 5*r^5*s^4 + 3*r^6*s^6 + r^2*(-1+s^2) + r^3*(1+14*s^2) + r^4*(s^2 - 5*s^4))))) / (2*n^(3/2)*sqrt(Pi)), where r = 0.2079338501416944274..., s = 1.815065347470593612... are roots of the system of equations (2*r^2*s*(1 + r - r^2*s^2))/(1 - 2*r - 2*r^3*s^2 + r^4*s^4 + r^2*(1 - 2*s^2))^(3/2) = 1, 1/sqrt(-4*r^3*s^2 + (-1 + r + r^2*s^2)^2) = s. - Vaclav Kotesovec, Mar 07 2014
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 24*x^4 + 86*x^5 + 328*x^6 + ...
where g.f. A(x) satisfies:
(1) A(x) = 1 + x*(1 + x*A(x)^2) + x^2*(1 + 4*x*A(x)^2 + x^2*A(x)^4) + x^3*(1 + 9*x*A(x)^2 + 9*x^2*A(x)^4 + x^3*A(x)^6) + x^4*(1 + 16*x*A(x)^2 + 36*x^2*A(x)^4 + 16*x^3*A(x)^6 + x^4*A(x)^8) + ...;
(2) A(x) = 1/(1-x) + x^2*A(x)^2*(1+x)/(1-x)^3 + x^4*A(x)^4*(1+4*x+x^2)/(1-x)^5 + x^6*A(x)^6*(1+9*x+9*x^2+x^3)/(1-x)^7 + ...
MATHEMATICA
Table[Sum[Binomial[n+k, 2*k]*Binomial[n+1, n-2*k]/(n+1), {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2014 *)
PROG
(PARI) {a(n)=sum(k=0, n\2, binomial(n+k, 2*k)*binomial(n+1, n-2*k))/(n+1)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(A^2+x*O(x^n))^k))); polcoeff(A, n)}
(PARI) {a(n)=polcoeff((1/x)*serreverse(x*(1-x^2)^2/(sqrt((1-x^2)^3+x^2*(1+x^2)^2+x*O(x^n))+x*(1+x^2))), n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/sqrt((1-x-x^2*A^2)^2-4*x^3*A^2+x*O(x^n))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1-x*A^2)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^k*(A^2+x^2*O(x^n))^k))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*(A^2+x*O(x^n))^m*sum(k=0, n, binomial(m+k, k)^2*x^k))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*A^(2*m)/(1-x+x*O(x^n))^(2*m+1)*sum(k=0, m, binomial(m, k)^2*x^k))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3, (2*m)!/m!^2*x^(3*m)*A^(2*m)/(1-x-x^2*A^2+x*O(x^n))^(2*m+1))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 12 2011
STATUS
approved