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URL: https://oeis.org/A127927

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A127927
G.f. A(x) satisfies: [x^(2n)] A(x)/Catalan(x)^n = A001764(n) = C(3n,n)/(2n+1) and [x^(2n+1)] A(x)/Catalan(x)^n = A001764(n+1) for n>=0, where Catalan(x) is the g.f. of A000108.
2
1, 1, 3, 9, 31, 108, 391, 1431, 5319, 19926, 75252, 285750, 1090491, 4177774, 16060401, 61916977, 239307063, 926929746, 3597296770, 13984508500, 54448030092, 212282062488, 828673761978, 3238495227846, 12669206034339
OFFSET
0,3
COMMENTS
Main diagonal of triangle A062745: a(n) = A062745(n,n) (see formula given in A062745 by Emeric Deutsch).
LINKS
FORMULA
a(n) = C(2*n,n) - (-1)^(n-1)*Sum_{i=0..[(n-1)/2]} C(3*i,i)*C(i-n-1,n-1-2*i)/(2*i+1).
From Vaclav Kotesovec, May 01 2018: (Start)
Recurrence: 2*(n-1)*n*(2*n + 1)*(5*n - 6)*a(n) = (n-1)^2*(115*n^2 - 138*n + 56)*a(n-1) + 4*(n-2)*(n+1)*(2*n - 3)*(5*n - 11)*a(n-2) - 36*(n-2)*(2*n - 5)*(2*n - 3)*(5*n - 1)*a(n-3).
a(n) ~ 4^n / (phi^2 * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. (End)
MATHEMATICA
a[n_] := Binomial[2*n, n] - (-1)^(n-1)*Sum[ Binomial[3*k, k]*Binomial[k - n-1, n-1-2*k]/(2*k+1), {k, 0, Floor[(n-1)/2]}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 30 2018 *)
PROG
(PARI) {a(n)=binomial(2*n, n)+(-1)^n*sum(i=0, (n-1)\2, binomial(3*i, i) *binomial(i-n-1, n-1-2*i)/(2*i+1))}
(Magma) [1] cat [Binomial(2*n, n) - (-1)^(n-1)*(&+[Binomial(3*k, k)*Binomial(k-n - 1, n-2*k-1)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Apr 30 2018
CROSSREFS
Cf. A062745; A001764 (ternary trees), A000108 (Catalan).
Sequence in context: A225340 A148966 A279971 * A148967 A189429 A123222
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2007
STATUS
approved