A064340 -id:A064340 - cp's OEIS frontend

A067297 Convolution of C(2,2; n) := A064340(n) (generalized Catalan numbers) with itself.

1, 2, 9, 64, 584, 6144, 70576, 859520, 10909440, 142773760, 1913027840, 26115182592, 361936623616, 5079287545856, 72033971859456, 1030768222437376, 14864066521333760, 215791593346695168

Offset: 0

Wolfdieter Lang, Feb 05 2002

Bisections: a(2*k) = A067320(k), a(2*k+1) = 2*A067321(k), k>=0.

a(n) = Sum_{k=0..n} C(2, 2; k)*C(2, 2; n-k).

G.f.: ((3+c(4*x))/(2*(1-2*x*c(4*x))*(1+c(4*x))))^2, with c(x) g.f. for A000108 (Catalan). Also: (c(4*x)*(3+c(4*x)))^2/(1+c(4*x))^4, or (-1+36*x +(1+24*x)*c(4*x))/((1-4*x)*(1+20*x)*c(4*x)-1+44*x-16*x^2).

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

A067327 Triangle related to generalized Catalan numbers A064340.

Original entry on oeis.org

1, 1, 3, 4, 12, 12, 28, 84, 96, 48, 256, 768, 912, 576, 192, 2704, 8112, 9792, 6720, 3072, 768, 31168, 93504, 113856, 81408, 42240, 15360, 3072, 380608, 1141824, 1397760, 1023744, 568320, 242688, 73728

Offset: 0

Wolfdieter Lang, Feb 05 2002

The row polynomials Z(2,2; n,y)= sum(a(n,m)*y^m,m=0..n) appear in c(2,2; x) (the g.f. of C(2,2; n) := A064340(n)) with the first (n+1) expansion terms subtracted, as follows: c(2,2; x)-sum(C(2,2; k)*x^k,k=0..n) = x^(n+1)*G(2,2; x)*Z(2,2; n,y), n>=0, where y=c(4*x) and c(x) is the g.f. of A000108 (Catalan) and G(2,2; x) is the g.f. of C(2,2; n+1), that is G(2,2; x)= (c(2,2; x)-1)/x. Hence G(2,2; x)*Z(2,2; k,c(4*x)) is, for k=0,1,..., the g.f. for C(2,2; n+k), n>=0.

Column sequences are: A064340(n), 3*A064340(n+1), Main diagonal gives A002001(n). Row sums give C(2,2; n+1)= A064340(n+1).

Cf. A067328 (scaled triangle with 1's in main diagonal).

a(n, 0)= C(2, 2; n) := A064340(n), n>=0; a(n, 1)= 3*C(2, 2; n), n>=1; a(n, m)=4*sum(a(n-1, k), k=(m-1)..(n-1)) if n>=m>=2, else 0.

A064879 Triangle of numbers composed of certain generalized Catalan numbers.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 5, 4, 1, 1, 0, 14, 28, 6, 1, 1, 0, 42, 256, 81, 8, 1, 1, 0, 132, 2704, 1566, 176, 10, 1, 1, 0, 429, 31168, 36126, 5888, 325, 12, 1, 1, 0, 1430, 380608, 921456, 238848, 16750, 540, 14, 1

Offset: 0

Wolfdieter Lang, Oct 12 2001

The column sequences (without leading zeros) for m=0..10 give: A019590, A000108, A064340-7, A064878. Row sums give A064880.

The sequence for column m (m >= 1) (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=:Y_{N}(N+1), N >=0, for alpha = beta = m. In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1. See also Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.

			{1}; {1,1}; {0,1,1}; {0,2,1,1}; {0,5,4,1,1}; ...

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.

W. Lang: First 10 rows.

a(n, m) = C(m, m; n-m) if n >= m, else 0, with C(m, m; n) := ((m^(2*(n-1)))/(n-1))*sum((k+1)*(k+2)*binomial(2*(n-2)-k, n-2-k)*((1/m)^(k+1)), k=0..n-2), n >= 2; C(m, m; 0) := 1=:C(m, m; 1).
G.f.: (x^m)*(1+(1-2*m)*x*c(x*m^2))/(1-m*x*c(x*m^2))^2 = (x^m)*((2*m-1)*c(x*m^2)*(m*x)^2 +(1-m)*(1-m+(1-3*m)*x))/(1-m-m*x)^2, m >= 0. For m >= 1 also: (x^m)*c(x*m^2)*(2*m-1+c(x*m^2)*(m-1)^2)/(1+(m-1)*c(x*m^2))^2.
In the G.f. the g.f. c(x) of A000108 (Catalan) appears.

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

A064341 Generalized Catalan numbers C(3,3; n).

Original entry on oeis.org

1, 1, 6, 81, 1566, 36126, 921456, 25055001, 711951606, 20891575566, 628237506276, 19259213633226, 599654171202156, 18911332670183856, 602840023457208516, 19392890824608619401, 628769286622411762086

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

Cf. A064340.

a(n) = ((9^(n-1))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/3)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.: (1-5*x*c(9*x))/(1-3*x*c(9*x))^2 = c(9*x)*(5+4*c(9*x))/(1+2*c(9*x))^2 = (5*c(9*x)*(3*x)^2+4*(1+4*x))/(2+3*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
2*(-n+1)*a(n) +3*(23*n-60)*a(n-1) +54*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A067305 Second column of triangle A067304.

Original entry on oeis.org

1, 5, 36, 328, 3440, 39408, 478912, 6068480, 79315200, 1061628160, 14480086016, 200540018688, 2812618092544, 39867889037312, 570237130752000, 8219880968028160, 119293333282291712, 1741605394647416832

Offset: 0

Wolfdieter Lang, Feb 05 2002

a(n)= A067304(n+1, 1) = A067297(n+1) - A064340(n+1), n>=0.

G.f.: 4*(3+c(4*x))*(c(4*x)^3)/(1+c(4*x))^4 with c(x) g.f. of A000108 (Catalan).

A067306 One-fourth of third column of triangle A067304.

Original entry on oeis.org

1, 8, 75, 796, 9176, 111936, 1421968, 18618560, 249542400, 3407171584, 47226230528, 662805371904, 9400304896000, 134517761982464, 1939837469085696, 28162286932246528, 411276783645753344

Offset: 0

Wolfdieter Lang, Feb 05 2002

a(n)= A067304(n+2, 2)/4 = (A067297(n+2) - (A064340(n+2)+A064340(n+1)))/4, n>=0.

G.f.: (3+10c(4*x)+3*c(4*x)^2)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).

A067307 One-fourth of fourth column of triangle A067304.

Original entry on oeis.org

7, 71, 768, 8920, 109232, 1390800, 18237952, 244701440, 3343713024, 46374830848, 651170275328, 9238908291072, 132251092529152, 1907671386263552, 27701755840561152, 404632598092447744

Offset: 0

Wolfdieter Lang, Feb 05 2002

a(n)= A067304(n+3, 3)/4 = (A067297(n+3)-(b(n+3)+b(n+2)+4*b(n+1)))/4, n>=0, with b(n) := A064340(n).

G.f.: 4(3+10*c(4*x)+12*c(4*x)^2+3*c(4*x)^3)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).

A067308 One sixteenth of fifth column of triangle A067304.

Original entry on oeis.org

16, 185, 2181, 26860, 342968, 4504944, 60509296, 827456576, 11482655232, 161302619392, 2289365653760, 32780329073664, 472951175022592, 6869148315201536, 100352220112662528, 1473672361011920896

Offset: 0

Wolfdieter Lang, Feb 05 2002

			21+70*y+93*y^2+60*y^3 = p(4,y), fifth row polynomial of triangle A067329.

a(n)= A067304(n+4, 4)/15 = (A067297(n+4)-sum(b(j)b(n+4-j), j=0..3))/16, n>=0, with b(n) := A064340(n).

G.f.: (21+70c(4*x)+93*c(4*x)^2+60*c(4*x)^3+12*c(4*x)^4)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).

cp's OEIS Frontend

A067297 Convolution of C(2,2; n) := A064340(n) (generalized Catalan numbers) with itself.

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

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A067327 Triangle related to generalized Catalan numbers A064340.

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A064879 Triangle of numbers composed of certain generalized Catalan numbers.

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A067299 Second column of triangle A067298.

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A064341 Generalized Catalan numbers C(3,3; n).

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A067305 Second column of triangle A067304.

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A067306 One-fourth of third column of triangle A067304.

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A067307 One-fourth of fourth column of triangle A067304.

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A067308 One sixteenth of fifth column of triangle A067304.

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