A067300 -id:A067300 - 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

A067299 Second column of triangle A067298.

Original entry on oeis.org

2, 5, 32, 284, 2960, 33872, 411776, 5221568, 68299520, 914858240, 12486496256, 173031701504, 2428066058240, 34432752275456, 492697174507520, 7104716644990976, 103142445617709056, 1506248913346691072

Offset: 0

Wolfdieter Lang, Feb 05 2002

Cf. A064340 (first column), A067300 (third column).

a(n)=A067298(n+1, 1).
G.f.: ((1+x)*c(2, 2; x)-1)/x, with c(2, 2; x) := (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 g.f. for A064340 and c(x) is g.f. for A000108 (Catalan).
G.f.: (2+x + 12*x*(1+x)*c(4*x))/(1+2*x)^2, where c(x) is g.f. for A000108 (Catalan). - Wolfdieter Lang, May 05 2006.
Conjecture: n*(17*n-7)*a(n) +2*(-119*n^2+270*n-96)*a(n-1) -16*(17*n+10)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jul 21 2016

A067301 One eighth of fourth column of triangle A067298.

Original entry on oeis.org

8, 41, 398, 4460, 53720, 677744, 8836832, 118109888, 1609484672, 22276767488, 312305704448, 4425515174912, 63285369657344, 912105965121536, 13235652537933824, 193215065765888000, 2835518953336438784

Offset: 0

Wolfdieter Lang, Feb 05 2002

Cf. A000108, A067298.

Cf. 3*A067300 (third column).

a(n) = A067298(n+3, 3)/8.

G.f.: ((-3+37*x)+(3+15*x-84*x^2)*c(4*x))/(8*x*(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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A067299 Second column of triangle A067298.

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A067301 One eighth of fourth column of triangle A067298.

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