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 A063902 a(n+1) = sum{j = 0,...n}[C(2n,2j)a(j)a(n-j)] with a(0) = 1. 4
 1, 1, 2, 10, 80, 1000, 17600, 418000, 12848000, 496672000, 23576960000, 1348404640000, 91442700800000, 7255463564800000, 665885747225600000, 69994087116448000000, 8354181454767104000000, 1123646013779238400000000, 169165728883243642880000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..270 Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014. FORMULA E.g.f. satisfies: A(x) = exp( Integral 1/A(x) * Integral A(x)^2 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(1*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015 From Vaclav Kotesovec, Jun 14 2015: (Start) a(n) ~ c * d^n * n!^2 * sqrt(n), where d = A258895 = 32*Pi / (Gamma(1/6) * Gamma(1/3))^2 = 2^(17/3) * Pi^2 / (3*Gamma(1/3)^6) = 0.452104299183420528841..., c = 1.53043521765866544548745...  = 2^(20/3) * Pi^(3/2) / Gamma(1/3)^6 = 192*sqrt(Pi) / (Gamma(1/3)*Gamma(1/6))^2. a(n) ~ 3 * 2^(5*n+7) * Pi^(n+3/2) * n^(2*n+3/2) / (exp(2*n) * Gamma(1/6)^(2*n+2) * Gamma(1/3)^(2*n+2)). a(n) ~ 2^((17*n+23)/3) * Pi^(2*n+5/2) * n^(2*n+3/2) / (3^n * exp(2*n) * Gamma(1/3)^(6*n+6)). (End) EXAMPLE a(3) = 1*a(0)a(2) + 6*a(1)a(1) + 1*a(2)a(0) = 2+6+2 = 10. E.g.f.: A(x) = 1 + x^2/2! + 2*x^4/4! + 10*x^6/6! + 80*x^8/8! + ... G.f. = 1 + x + 2*x^2 + 10*x^3 + 80*x^4 + 1000*x^8 + 17600*x^9 + ... MATHEMATICA Clear[a]; a[n_]:=a[n]=If[n<2, 1, Sum[a[j]*a[n-1-j]*Binomial[2*n-2, 2*j], {j, 0, n-1}]]; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Jun 14 2015 *) PROG (PARI) /* E.g.f. A(x) = exp( Integral 1/A(x) * Integral A(x)^2 dx dx ) */ {a(n) = local(A=1+x); for(i=1, n, A = exp( intformal( 1/A * intformal( A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A, n)} for(n=0, 20, print1(a(2*n), ", ")) \\ Paul D. Hanna, Jun 02 2015 (PARI) /* By definition: */ {a(n) = if(n==0, 1, sum(k=0, n-1, binomial(2*n-2, 2*k)*a(k)*a(n-k-1)))} for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 02 2015 CROSSREFS a(n+1) = sum[C(n, j)a(j)a(n-j)] would give factorials A000142, a(n+1) = sum[C(2n, 2j)a(j)a(n-j)]/a(n) would give Catalan numbers A000108, a(n+1) = sum[C(n, j)a(j)a(n-j)]/a(n) would give central binomials A001405. Cf. A007558, A258895. Sequence in context: A227464 A269353 A064312 * A088351 A231919 A174962 Adjacent sequences:  A063899 A063900 A063901 * A063903 A063904 A063905 KEYWORD nonn AUTHOR Henry Bottomley, Aug 30 2001 EXTENSIONS More terms from Vaclav Kotesovec, Jun 14 2015 STATUS approved

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Last modified October 20 02:18 EDT 2019. Contains 328244 sequences. (Running on oeis4.)