A067301 -id:A067301 - cp's OEIS frontend

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

Wolfdieter Lang, Feb 05 2002

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.

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

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.

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

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, ...

More terms from Jinyuan Wang, Apr 20 2025

A067300 One third of third column of triangle A067298.

Original entry on oeis.org

3, 12, 100, 1024, 11632, 140864, 1782080, 23273984, 311407360, 4246776832, 58812433408, 824868814848, 11692779565056, 167254617440256, 2411126992093184, 34994856661483520, 510941885186637824

Offset: 0

Wolfdieter Lang, Feb 05 2002

Cf. A067299 (second column), 8*A067301 (fourth column).

a(n)=A067298(n+2, 2)/3.

G.f.: ((1-4*x)*c(4*x)+2)/(1-2*x*c(4*x))^2, with c(x) is g.f. for A000108 (Catalan).

cp's OEIS Frontend

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

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