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 A153299 G.f.: A(x) = F(x*G(x)) where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan). 2
 1, 1, 4, 20, 111, 657, 4067, 26028, 170913, 1145446, 7804797, 53911104, 376669462, 2657391772, 18904566514, 135460704648, 976795422828, 7082951967141, 51614974500605, 377798933519164, 2776363089297553, 20476554379564305 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(2n-k,n-k)*k/(2n-k) for n>0 with a(0)=1. G.f. satisfies: A(x) = 1 + x*G(x)*A(x)^3 where G(x) is the g.f. of A000108. G.f. satisfies: A(x*F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A001764. EXAMPLE G.f.: A(x) = F(x*G(x)) = 1 + x + 4*x^2 + 20*x^3 + 111*x^4 +... Related expansions. F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +... F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +... G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +... G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 +... A(x)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 278*x^4 + 1696*x^5 +... A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 513*x^4 + 3225*x^5 +... G(x)*A(x)^3 = 1 + 4*x + 20*x^2 + 111*x^3 + 657*x^4 +... PROG (PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*binomial(2*(n-k)+k, n-k)*k/(2*(n-k)+k)))} CROSSREFS Cf. A000108, A001764; A153298, A153390. Sequence in context: A026156 A025183 A014523 * A239643 A081335 A136783 Adjacent sequences:  A153296 A153297 A153298 * A153300 A153301 A153302 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 15 2009 STATUS approved

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Last modified December 6 01:25 EST 2021. Contains 349558 sequences. (Running on oeis4.)