A067298 -id:A067298 - cp's OEIS frontend

A067304 Generalized Catalan triangle A067298 with row reversion.

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

A067302 Row sums of triangle A067298 and of A067304.

Original entry on oeis.org

1, 3, 18, 160, 1752, 21504, 282304, 3867840, 54547200, 785255680, 11478167040, 169748686848, 2533556365312, 38094656593920, 576271774875648, 8761529890717696, 133776598692003840, 2050020136793604096

Offset: 0

Wolfdieter Lang, Feb 05 2002

a(n)=sum(A067298(n, m), m=0..n ).
Bisection: a(2*k)= (k+1)*A067297(2*k)=: A067303(k), a(2*k+1)= (2*k+3)*A067297(2*k+1)/2 =: A067322(k), k>=0.
G.f.: ge(x^2) + x*go(x^2) with ge(x) g.f. of A067303 and go(x) g.f. of A067322.

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).

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).

A064340 Generalized Catalan numbers C(2,2; n).

Original entry on oeis.org

1, 1, 4, 28, 256, 2704, 31168, 380608, 4840960, 63458560, 851399680, 11635096576, 161396604928, 2266669453312, 32166082822144, 460531091685376, 6644185553305600, 96498260064403456, 1409750653282287616, 20702370737659052032, 305428492830594039808

Offset: 0

Wolfdieter Lang, Oct 12 2001

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.

Cf. A000108 (Catalan as C(1,1; n)), A064879, A067298.

my(x='x+O('x^30)); Vec((1+(13-3*sqrt(1-16*x))*x/2)/(1+2*x)^2) \\ Jinyuan Wang, Apr 20 2025

a(n) = ((4^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1), n >= 2, a(0) = a(1) = 1.
G.f.: (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 = c(4*x)*(3+c(4*x))/(1+c(4*x))^2 = (1+5*x+3*c(4*x)*(2*x)^2)/(1+2*x)^2 with c(x) = A(x) g.f. of Catalan numbers A000108.
(-n+1)*a(n) + 2*(7*n-20)*a(n-1) + 16*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Aug 09 2017

cp's OEIS Frontend

A067304 Generalized Catalan triangle A067298 with row reversion.

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A067302 Row sums of triangle A067298 and of A067304.

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A067300 One third of third column of triangle A067298.

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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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A064340 Generalized Catalan numbers C(2,2; n).

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