A067302 -id:A067302 - cp's OEIS frontend

A067303 Even-indexed members of A067302.

1, 18, 1752, 282304, 54547200, 11478167040, 2533556365312, 576271774875648, 133776598692003840, 31513560479297044480, 7505638177922587557888, 1802924878727702252617728, 436026430783762289982963712

Offset: 0

Wolfdieter Lang, Feb 05 2002

Cf. A067322 (odd-indexed members of A067302).

a(n) = A067302(2n) = (n+1)*A067297(2*n).

G.f.: (x*(d/dx)A(x) + A(x)) with A(x) g.f. of A067320.

A067322 Odd-indexed members of A067302.

Original entry on oeis.org

3, 160, 21504, 3867840, 785255680, 169748686848, 38094656593920, 8761529890717696, 2050020136793604096, 485760548730263568384, 116217935644216514314240, 28016590108689074277580800

Offset: 0

Wolfdieter Lang, Feb 05 2002

Cf. A067303 (even-indexed members of A067302).

a(n) = A067302(2n+1) = (2*n+3)*A067297(2*n+1)/2.

G.f.: A(x) = (2*x*(d/dx)g(x) + 3*g(x)) with g(x) g.f. of A067321.

A067304 Generalized Catalan triangle A067298 with row reversion.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Feb 05 2002

Identity for each row n >= 1: T(n, m) + T(n, n-m+1) = A067297(n+1) (convolution of generalized Catalan numbers) for every m = 1..floor((n+1)/2). E.g., T(2*k+1, k+1) = A067297(2*(k+1))/2.

			Triangle begins:
    1;
    2,   1;
    9,   5,   4;
   64,  36,  32,  28;
  584, 328, 300, 284, 256;
  ...
n=3: T(3, 0) = 64 = 36+28 = 32+32.

The columns give for m=0..4: A067297 (diagonals of A067298), A067305, A067306, A067307, A067308.
Cf. A067302 (row sums), A067323 (corresponding triangle for ordinary Catalan numbers).
Cf. A000108, A067298, A067329.

T(n, m) = A067298(n, n-m), n >= m >= 0, otherwise 0.

G.f. for column m >= 1 (without leading zeros): (2^(2*ceiling(m/2))*p(m, y)*(y^3)/(1+y)^4, where y = y(x) = c(4*x), with c(x) = g.f. of A000108 (Catalan) and the row polynomials p(n, y) = Sum_{k=0..n} A067329(n, k)*y^k, n >= 1. For m = 0: ((y*(3+y))^2)/(1+y)^4 with y = y(x) = c(4*x) (see A067297).

More terms from Jinyuan Wang, Apr 20 2025

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

Original entry on oeis.org

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

cp's OEIS Frontend

A067303 Even-indexed members of A067302.

Views

Author

Keywords

Crossrefs

Formula

A067322 Odd-indexed members of A067302.

Views

Author

Keywords

Crossrefs

Formula

A067304 Generalized Catalan triangle A067298 with row reversion.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions