A064094 -id:A064094 - cp's OEIS frontend

A064088 Generalized Catalan numbers C(5; n).

1, 1, 6, 61, 766, 10746, 161376, 2537781, 41260086, 687927166, 11698135396, 202104763026, 3537486504556, 62595852983236, 1117926476207316, 20124876291104421, 364797768048805926, 6652740911381353206, 121975721251036497636, 2247064873245590484966, 41573071647518070152196

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=5, beta =1 (or alpha=1, beta=5).

G. C. Greubel, Table of n, a(n) for n = 0..850

Cf. A064087 (C(4, n)).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9-Sqrt(1-20*x))/(2*(x+4)) )); // G. C. Greubel, May 02 2019

a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*5^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(9-Sqrt[1-20*x])/(2*(x+4)), {x,0,30}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-4*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* Ralf Stephan */

( (9-sqrt(1-20*x))/(2*(x+4)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1+5*x*c(5*x)/4)/(1+x/4) = 1/(1-x*c(5*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(5^m)/n, n >= 1, a(0) := 1.
a(n) = (-1/4)^n*(1 - 5*Sum_{k=0..n-1} C(k)*(-20)^k); with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*5^(n-k). - Philippe Deléham, Jan 19 2004
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  5, 5, 5, 0, 0, 0, ...
  5, 5, 5, 5, 0, 0, ...
  5, 5, 5, 5, 5, 0, ...
  5, 5, 5, 5, 5, 5, ...
  ... (End)
Conjecture: 4*n*a(n) +(-79*n+120)*a(n-1) +10*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 5^(n+1) / (81*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064089 Generalized Catalan numbers C(6; n).

Original entry on oeis.org

1, 1, 7, 85, 1279, 21517, 387607, 7312789, 142648495, 2853691357, 58226571271, 1207062556261, 25351452769567, 538285926177325, 11535690316148215, 249189167966657845, 5420206822556721295

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=6, beta =1 (or alpha=1, beta=6).

G. C. Greubel, Table of n, a(n) for n = 0..725

Cf. A064088 (C(5, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (11 -Sqrt(1-24*x))/(2*(x+5)) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[(11 -Sqrt[1-24*x])/(2*(x+5)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-5*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((11 -sqrt(1-24*x))/(2*(x+5))) \\ G. C. Greubel, May 02 2019

((11 -sqrt(1-24*x))/(2*(x+5))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 6*x*c(6*x)/5)/(1+x/5) = 1/(1 - x*c(6*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(6^m)/n.
a(n) = (-1/5)^n*(1 - 6*Sum_{k=0..n-1} C(k)*(-30)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*6^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 5*n*a(n) +(-119*n+180)*a(n-1) +12*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 2^(3*n + 1) * 3^(n+1) / (121*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A113647 Triangle of numbers related to the generalized Catalan sequence C(2;n+1)=A064062(n+1), n>=0.

Original entry on oeis.org

1, 1, 3, 1, 7, 13, 1, 15, 41, 67, 1, 31, 113, 247, 381, 1, 63, 289, 783, 1545, 2307, 1, 127, 705, 2271, 5361, 9975, 14589, 1, 255, 1665, 6207, 16929, 36879, 66057, 95235, 1, 511, 3841, 16255, 50113, 123871, 255985, 446455, 636925, 1, 1023, 8705, 41215, 141441

Offset: 0

Wolfdieter Lang, Jan 13 2006

This triangle, called Y(2,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=2, beta=1. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(2,n):=A064062(n).
The diagonal sequences give A064062(n+1), A115137, A115150-A115153, for n+1>= M=1,..,6.

			Triangle begins:
  1;
  1,3;
  1,7,13;
  1,15,41,67;
  1,31,113,247,381;
  ...
113=a(4,3)= a(4,2) + 2*a(3,3)= 31 + 2*41.

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.
Wolfdieter Lang, First 10 rows.

Row sums give A115136.

a(n, n+1)=A064062(n+1) (main diagonal with M=1); a(n, n-M+2)= a(n, n-M+1) + 2*a(n-1, n-M+2), M>=2; a(n, 1)=1; n>=0.
G.f. for diagonal sequence M=1: GY(1, x):=(2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108 (Catalan); for M=2: GY(2, x)=(1-2*x)*GY(1, x)-1; for M>=3: GY(M, x)= GY(M-1, x) -2*x*GY(M-2, x) + x^(M-2).
G.f. for diagonal sequence M (solution to the above given recurrence): GY(M, x)= (x^(M-1)/(1+x))*( 2^(M+1)*x*(p(M, 2*x)-(2*x)*p(M+1, 2*x)*c(2*x))+1), with c(x) g.f. of A000108 (Catalan) and p(n, x):= -((1/sqrt(x))^(n+1))*S(n-1, 1/sqrt(x)) with Chebyshev's S(n, x) polynomials given in A049310.

A064090 Generalized Catalan numbers C(7; n).

Original entry on oeis.org

1, 1, 8, 113, 1982, 38886, 817062, 17981769, 409186310, 9549411950, 227307541448, 5497312072330, 134696099554276, 3336563455537768, 83419226227330722, 2102274863070771033, 53347639317495439302

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=7, beta =1 (or alpha=1, beta=7).

G. C. Greubel, Table of n, a(n) for n = 0..690

Cf. A064089 (C(6, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (13-Sqrt(1-28*x))/(2*(x+6)) )); // G. C. Greubel, May 02 2019

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*7^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(13 -Sqrt[1-28*x])/(2*(x+6)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-6*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((13-sqrt(1-28*x))/(2*(x+6))) \\ G. C. Greubel, May 02 2019

((13-sqrt(1-28*x))/(2*(x+6))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1+7*x*c(7*x)/6)/(1+x/6) = 1/(1-x*c(7*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(7^m)/n.
a(n) = (-1/6)^n*(1 - 7*Sum_{k=0..n-1} C(k)*(-42)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*7^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 6*n*a(n) +(-167*n+252)*a(n-1) +14*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 7^(n+1) / (169*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064091 Generalized Catalan numbers C(8; n).

Original entry on oeis.org

1, 1, 9, 145, 2905, 65121, 1563561, 39322929, 1022586105, 27272680705, 741894295369, 20504949587409, 574176887116441, 16254518495907745, 464436319229036265, 13376293681432402545, 387925710986712480825

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=8, beta =1 (or alpha=1, beta=8).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A064090 (C(7, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (15 - Sqrt(1-32*x))/(2*(x+7)) )); // G. C. Greubel, May 02 2019

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n+m-1, m]*(8^m)/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 21 2013 *)
Table[FullSimplify[(-1)^(2*n)*2^(3+5*n)*(1/2*(2*n-1))! Hypergeometric2F1[1,1/2+n,2+n,-224]/(Sqrt[Pi]*(n+1)!)],{n,0,20}] (* Vaclav Kotesovec, Aug 13 2013 *)
CoefficientList[Series[(15 -Sqrt[1-32*x])/(2*(x+7)), {x,0,20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-7*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((15 -sqrt(1-32*x))/(2*(x+7))) \\ G. C. Greubel, May 02 2019

((15 -sqrt(1-32*x))/(2*(x+7))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 8*x*c(8*x)/7)/(1+x/7) = 1/(1 - x*c(8*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(8^m)/n.
a(n) = (-1/7)^n*(1 - 8*Sum_{k=0..n-1} C(k)*(-56)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*8^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 7*n*a(n) +(-223*n+336)*a(n-1) +16*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 2^(5*n+3)/(225*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

A064092 Generalized Catalan numbers C(9; n).

Original entry on oeis.org

1, 1, 10, 181, 4078, 102826, 2777212, 78571837, 2298558934, 68964092542, 2110472708140, 65620725560578, 2067160250751436, 65833929303952564, 2116166898185821792, 68565914052628406221, 2237022199842087256678

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=9, beta =1 (or alpha=1, beta=9).

G. C. Greubel, Table of n, a(n) for n = 0..640

Cf. A064091 (C(8, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (17 - Sqrt(1-36*x))/(2*(x+8)) )); // G. C. Greubel, May 02 2019

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*9^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(17 -Sqrt[1-36*x])/(2*(x+8)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-8*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

my(x='x+O('x^20)); Vec((17 -sqrt(1-36*x))/(2*(x+8))) \\ G. C. Greubel, May 02 2019

((17 -sqrt(1-36*x))/(2*(x+8))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 9*x*c(9*x)/8)/(1+x/8) = 1/(1 - x*c(9*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(9^m)/n.
a(n) = (-1/8)^n*(1 - 9*Sum_{k=0..n-1} C(k)*(-72)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*9^(n-k). - Philippe Deléham, Jan 19 2004
Conjecture: 8*n*a(n) +(-287*n+432)*a(n-1) +18*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 9^(n+1) / (289*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A064093 Generalized Catalan numbers C(10; n).

Original entry on oeis.org

1, 1, 11, 221, 5531, 154941, 4649451, 146150061, 4750427771, 158361063581, 5384626548491, 186023930383501, 6511108452179611, 230400987949757821, 8228844334672249131, 296245683962814194541, 10739133812893020645051

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=10, beta =1 (or alpha=1, beta=10).

In general, for m>=1, C(m; n) ~ m * (4*m)^n / ((2*m - 1)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

G. C. Greubel, Table of n, a(n) for n = 0..620

Cf. A064092 (C(9, n)).

Cf. A000108, A059365.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (19 - Sqrt(1-40*x))/(2*(x+9)) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[(19 -Sqrt[1-40*x])/(2*(x+9)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

my(x='x+O('x^20)); Vec((19 -sqrt(1-40*x))/(2*(x+9))) \\ G. C. Greubel, May 02 2019

((19 -sqrt(1-40*x))/(2*(x+9))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

G.f.: (1 + 10*x*c(10*x)/9)/(1+x/9) = 1/(1 - x*c(10*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(10^m)/n.
a(n) = (-1/9)^n*(1 - 10*Sum_{k=0..n-1} C(k)*(-90)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*10^(n-k). - Philippe Deléham, Jan 19 2004
a(n) ~ 2^(3*n + 1) * 5^(n+1) / (361*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 5, 20, 28, 14, 0, 14, 70, 135, 120, 42, 0, 42, 252, 616, 770, 495, 132, 0, 132, 924, 2730, 4368, 4004, 2002, 429, 0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430, 0, 1430, 12870, 51051, 116688, 168300, 157080, 92820, 31824, 4862

Offset: 0

Philippe Deléham, Jun 03 2004, Jun 14 2007

			Triangle begins:
  1;
  0,   1;
  0,   1,    2;
  0,   2,    6,     5;
  0,   5,   20,    28,    14;
  0,  14,   70,   135,   120,    42;
  0,  42,  252,   616,   770,   495,   132;
  0, 132,  924,  2730,  4368,  4004,  2002,  429;
  0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
B. Derrida, E. Domany and D. Mukamel, A exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs.(20), (21), p. 672.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Variant of A062991, unsigned and transposed.
See also A234950 for another version.
Columns: A000007 (k=0), 2*A001700 (k=1).
Diagonals: A002694 (k=n-1), A000108 (k=n).
Row sums: A064062 (generalized Catalan C(2; n)).
Cf. A000012, A002694, A064062, A064063, A064087, A064088, A064089.
Cf. A064090, A064091, A064092, A064093, A064094.

A094385:= func< n,k | n eq 0 select 1 else Binomial(2*n, k-1)*Binomial(2*n-k-1, n-k)/n >;
[A094385(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 26 2024

T[n_, k_] := Binomial[2n, k-1] Binomial[2n-k-1, n-k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

def A094385(n,k): return 1 if (n==0) else binomial(2*n,k-1)*binomial(2*n-k-1, n-k)//n
flatten([[A094385(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 26 2024

T is given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.
Sum_{k = 0..n} T(n, k)*x^(n-k) = C(x+1; n), generalized Catalan numbers; see left diagonals of triangle A064094: A000012, A000108, A064062, A064063, A064087..A064093 for x = -1, 0, ..., 9, respectively.
From G. C. Greubel, Sep 26 2024: (Start)
T(n, 1) = A000108(n-1), n >= 1.
T(n, n-1) = A002694(n), n >= 1.
T(n, n) = A000108(n). (End)

New name using a formula of the author by Peter Luschny, Sep 26 2024

A115154 Triangle of numbers related to the generalized Catalan sequence C(3;n+1) = A064063(n+1), n>=0.

Original entry on oeis.org

1, 1, 4, 1, 13, 25, 1, 40, 115, 190, 1, 121, 466, 1036, 1606, 1, 364, 1762, 4870, 9688, 14506, 1, 1093, 6379, 20989, 50053, 93571, 137089, 1, 3280, 22417, 85384, 235543, 516256, 927523, 1338790, 1, 9841, 77092, 333244, 1039873, 2588641, 5371210

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle, called Y(3,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=3, beta=1. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(3,n+1):=A064063(n+1).
The diagonal sequences give A064063(n+1), A115188-A115192 for n+1>= M=1,..,6.

			Triangle begins:
  1;
  1,   4;
  1,  13,  25;
  1,  40, 115,  190;
  1, 121, 466, 1036, 1606;
  ...
466 = a(4,3) = a(4,2) + 3*a(3,3) = 121 + 3*115.

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.
Wolfdieter Lang, First 10 rows.

Row sums give A115187.

a(n,n+1)=A064063(n+1) (main diagonal with M=1); a(n,n-M+2)= a(n,n-M+1) + 3*a(n-1,n-M+2), M>=2; a(n,1)=1; n>=0.
G.f. for diagonal sequence M=1: GY(1,x):=(3*c(3*x)-1)/(2+x) with c(x) the o.g.f. of A000108 (Catalan); for M=2: GY(2,x)=(1-3*x)*GY(1,x)-1; for M>=3: GY(M,x)= GY(M-1,x) - 3*x*GY(M-2,x) + 2*x^(M-2).
G.f. for diagonal sequence M (solution to the above given recurrence): GY(M,x)= (x^(M-1)/(1+x))*( 3^(M+1)*x*(p(M,3*x)-(3*x)*p(M+1,3*x)*c(3*x))+1), with c(x) g.f. of A000108 (Catalan) and p(n,x):= -((1/sqrt(x))^(n+1))*S(n-1,1/sqrt(x)) with Chebyshev's S(n,x) polynomials given in A049310.

A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

Original entry on oeis.org

1, 2, 3, 4, 10, 13, 8, 28, 54, 67, 16, 72, 180, 314, 381, 32, 176, 536, 1164, 1926, 2307, 64, 416, 1488, 3816, 7668, 12282, 14589, 128, 960, 3936, 11568, 26904, 51468, 80646, 95235, 256, 2176, 10048, 33184, 86992, 189928, 351220, 541690, 636925, 512, 4864

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle Y(1,2) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=2. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(2,n+1):=A064062(n+1).
The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.

			Triangle begins:
   1;
   2,  3;
   4, 10,  13;
   8, 28,  54,  67;
  16, 72, 180, 314, 381;
  ...

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.
Wolfdieter Lang, First 10 rows.

Row sums give A084076.

G.f. m-th diagonal, m>=1: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^m)/(2*x*(1+x)) with c(x) the o.g.f. of A000108 (Catalan).

cp's OEIS Frontend

A064088 Generalized Catalan numbers C(5; n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A064089 Generalized Catalan numbers C(6; n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A113647 Triangle of numbers related to the generalized Catalan sequence C(2;n+1)=A064062(n+1), n>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A064090 Generalized Catalan numbers C(7; n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A064091 Generalized Catalan numbers C(8; n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A064092 Generalized Catalan numbers C(9; n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A064093 Generalized Catalan numbers C(10; n).

A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.