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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A067327 Triangle related to generalized Catalan numbers A064340.

Original entry on oeis.org

1, 1, 3, 4, 12, 12, 28, 84, 96, 48, 256, 768, 912, 576, 192, 2704, 8112, 9792, 6720, 3072, 768, 31168, 93504, 113856, 81408, 42240, 15360, 3072, 380608, 1141824, 1397760, 1023744, 568320, 242688, 73728
Offset: 0

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Author

Wolfdieter Lang, Feb 05 2002

Keywords

Comments

The row polynomials Z(2,2; n,y)= sum(a(n,m)*y^m,m=0..n) appear in c(2,2; x) (the g.f. of C(2,2; n) := A064340(n)) with the first (n+1) expansion terms subtracted, as follows: c(2,2; x)-sum(C(2,2; k)*x^k,k=0..n) = x^(n+1)*G(2,2; x)*Z(2,2; n,y), n>=0, where y=c(4*x) and c(x) is the g.f. of A000108 (Catalan) and G(2,2; x) is the g.f. of C(2,2; n+1), that is G(2,2; x)= (c(2,2; x)-1)/x. Hence G(2,2; x)*Z(2,2; k,c(4*x)) is, for k=0,1,..., the g.f. for C(2,2; n+k), n>=0.
Column sequences are: A064340(n), 3*A064340(n+1), Main diagonal gives A002001(n). Row sums give C(2,2; n+1)= A064340(n+1).

Crossrefs

Cf. A067328 (scaled triangle with 1's in main diagonal).

Formula

a(n, 0)= C(2, 2; n) := A064340(n), n>=0; a(n, 1)= 3*C(2, 2; n), n>=1; a(n, m)=4*sum(a(n-1, k), k=(m-1)..(n-1)) if n>=m>=2, else 0.
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