A067298 Generalized Catalan triangle, based on C(2,2; n) = A064340(n).

1, 1, 2, 4, 5, 9, 28, 32, 36, 64, 256, 284, 300, 328, 584, 2704, 2960, 3072, 3184, 3440, 6144, 31168, 33872, 34896, 35680, 36704, 39408, 70576, 380608, 411776, 422592, 429760, 436928, 447744, 478912, 859520, 4840960, 5221568, 5346240, 5421952, 5487488, 5563200, 5687872, 6068480, 10909440

Offset

0,3

Comments

For corresponding Catalan triangle with C(1,1; n) = A000108(n) see A028364.

Identity for each row n>=1: T(n, m) + T(n, n-(m+1)) = T(n, n) = A067297(n) for m=0..floor((n-1)/2). E.g., T(2*k+1, k) = A067297(2*k+1)/2.

Links

Table of n, a(n) for n=0..44.

Formula

T(n, m) = Sum_{i=0..m} C(2,2; i)*C(2,2; n-i) if n >= m >= 0 else 0.

G.f. for column m (without leading zeros): (c(m, x)*c(2,2; x)-c2(m-1, x))/x^m, with c(2,2; x) = (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 (g.f. for C(2,2; n)), c(x) = g.f. for Catalan numbers A000108, c(m, x) = Sum_{n=0..m} C(2,2; n)*x^n and c2(m, x) = Sum_{n=0..m} A067297(n)*x^n for m=0, 1, 2, ...

Example

Triangle begins:
    1;
    1,   2;
    4,   5,   9;
   28,  32,  36,  64;
  256, 284, 300, 328, 584;
  ...

Prog

(PARI) A064340(n) = if(n>1, sum(m=0, n-2, (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1))*(4^(n-1))/(n-1), 1);
T(n, m) = sum(i=0, m, A064340(i)*A064340(n-i)); \\ Jinyuan Wang, Apr 20 2025

Crossrefs

The columns (without leading zeros) give for m=0..3: A064340, A067299, 3*A067300, 8*A067301.

The main diagonal gives A067297. The row sums give A067302.

Cf. A000108, A067304.

Sequence in context: A073151 A279786 A255515 * A077389 A122991 A245512

Adjacent sequences: A067295 A067296 A067297 * A067299 A067300 A067301

Keyword

nonn,tabl

Author

Wolfdieter Lang, Feb 05 2002

Extensions

More terms from Jinyuan Wang, Apr 20 2025

Status

approved