A115193 -id:A115193 - cp's OEIS frontend

A115202 Fifth column of triangle A115193 (called C(1,2)).

1, 9, 67, 477, 3363, 23741, 168451, 1202685, 8641539, 62470141, 454164483, 3319054333, 24371503107, 179736723453, 1330803769347, 9889323810813, 73733148770307, 551423090098173, 4135500638060547

Offset: 0

Wolfdieter Lang, Feb 03 2006

Also one eighth of the fourth diagonal of triangle A115195, called Y(1,2).

G. C. Greubel, Table of n, a(n) for n = 0..500

f[n_] := SeriesCoefficient[(1 - 9*x + 14*x^2 - (1 - 5*x + 2*x^2) Sqrt[1 - 8*x])/(16*x^4*(1 + x)), {x, 0, n}];
Table[f[n], {n, 0, 50}] (* G. C. Greubel, Feb 04 2016 *)

a(n)= A115195(3+n,1+n)/8, n>=0.
G.f.: (-1 + 3*x + (1- 5*x + 2*x^2)*c(2*x))/(4*(1+x)*x^3), with the o.g.f. c(x) of A000108 (Catalan).
a(n) = A115193(4+n,4), n>=0.
a(n) = (-1)^n*8^(n+2)*(binomial(1/2, n+3)*Hypergeometric2F1(1,n+5/2; n+4; -8) + 20*binomial(1/2, n+4)*Hypergeometric2F1(1,n+7/2; n+5; -8) + 32*binomial(1/2, n+5)*Hypergeometric2F1(1,n+9/2; n+6; -8)). - G. C. Greubel, Feb 04 2016
D-finite with recurrence (n+4)*a(n) +2*(-6*n-13)*a(n-1) +(29*n-10)*a(n-2) +2*(13*n+22)*a(n-3) +8*(-2*n+3)*a(n-4)=0. - R. J. Mathar, Mar 10 2022

A115204 Seventh column of triangle A115193 (called C(1,2)).

Original entry on oeis.org

1, 13, 123, 1037, 8291, 64509, 494595, 3761661, 28486659, 215277565, 1625688067, 12277764093, 92783468547, 701828038653, 5314762113027, 40297495658493, 305941006516227, 2325794003091453, 17704219384479747

Offset: 0

Wolfdieter Lang, Feb 03 2006

Also sixth diagonal of triangle A115195, called Y(1,2), divided by 32.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A115193, A115195, A115202, A115203.

f[n_] := SeriesCoefficient[(1 - 13*x + 46*x^2 - 36*x^3 -(1 - 9*x + 18*x^2 - 4*x^3) Sqrt[1 - 8*x])/(64*x^6*(1 + x)), {x, 0, n}];
Table[f[n], {n, 0, 50}] (* G. C. Greubel, Feb 04 2016 *)

a(n) = A115195(5+n,1+n)/32, n>=0.
G.f.: (-1 + 7*x - 8*x^2 + (1- 9*x + 18*x^2 - 4*x^3)*c(2*x))/(16*(1+x)*x^5), with the o.g.f. c(x) of A000108 (Catalan).
G.f. is also: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^6)/(64*(1+x)*x^6).
a(n) = A115193(6+n,6), n>=0.
a(n) = (-1)^n*2^(8+3*n)*(Binomial[1/2, 4 + n]*Hypergeometric2F1[1, 7/2 + n, 5 + n, -8] + 4*(9*Binomial[1/2, 5 + n]*Hypergeometric2F1[1, 9/2 + n, 6 + n, -8] + 36*Binomial[1/2, 6 + n]*Hypergeometric2F1[1, 11/2 + n, 7 + n, -8] + 32*Binomial[1/2, 7 + n]*Hypergeometric2F1[1, 13/2 + n, 8 + n, -8])). - G. C. Greubel, Feb 04 2016
D-finite with recurrence 2*n*(n+6)*a(n) +(-11*n^2-51*n-120)*a(n-1) +(-37*n^2-99*n-132)*a(n-2) -12*(n+1)*(2*n+1)*a(n-3)=0. - R. J. Mathar, Mar 10 2022

A115203 Sixth column of triangle A115193 (called C(1,2)).

Original entry on oeis.org

1, 11, 93, 723, 5437, 40323, 297469, 2191875, 16164861, 119443459, 884719613, 6570430467, 48927031293, 365303660547, 2734459846653, 20518848036867, 154328140087293, 1163305103130627, 8787088644243453

Offset: 0

Wolfdieter Lang, Feb 03 2006

Also fifth diagonal of triangle A115195, called Y(1,2), divided by 16.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A115202, A115204.

f[n_] := SeriesCoefficient[(1 - 11*x + 28*x^2 - 8*x^3 - (1 - 7*x + 8*x^2) Sqrt[1 - 8*x])/(32*x^5*(1 + x)), {x, 0, n}];
Table[f[n], {n, 0, 50}] (* G. C. Greubel, Feb 04 2016 *)

a(n) = A115195(4+n,1+n)/16, n>=0.
G.f.: (-1 + 5*x -2*x^2 + (1- 7*x + 8*x^2)*c(2*x))/(8*(1+x)*x^4) with the o.g.f. c(x) of A000108 (Catalan).
G.f. is also: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^5)/(32*(1+x)*x^5).
a(n) = A115193(5+n,5), n>=0.
a(n) = (-1)^(n+1)* 2^(10 + 3*n)*(binomial(1/2,n+4)*Hypergeometric2F1(1, 7/2 + n, 5 + n, -8) + 7*binomial(1/2,n+5)*Hypergeometric2F1(1, 9/2 + n, 6 + n, -8) + 8*binomial(1/2,n+6)*Hypergeometric2F1(1, 11/2 + n, 7 + n, -8)). - G. C. Greubel, Feb 04 2016
a(n) ~ 2^(3*n+10) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 05 2016
D-finite with recurrence (n+5)*a(n) +2*(-7*n-22)*a(n-1) +(49*n+43)*a(n-2) +124*a(n-3) +32*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Mar 10 2022

A084076 Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.

Original entry on oeis.org

1, 5, 27, 157, 963, 6141, 40323, 270845, 1852419, 12857341, 90337283, 641286141, 4592533507, 33139654653, 240723001347, 1758796578813, 12916805074947, 95300512382973, 706044251602947, 5250379998560253, 39176121681444867

Offset: 0

Wouter Meeussen, May 11 2003

nonn

2*a(n-1) is the second diagonal of the triangle A115195.

Row sums of A167432. Hankel transform is A167435. - Paul Barry, Nov 03 2009

			{1}
{-2,-1,0,1,2}
{-3,-2,-1,0,1,2,3,-2,-1,0,1,2,-1,0,1,-2,-1,0,1,2,-3,-2,-1,0,1,2,3}

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

Third column (m=2) of triangle A115193, called C(1, 2).
Cf. A000108, A115139, A115195, A167432, A167435.
Cf. A084075, A084077, A084078.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-5*x -(1- x)*Sqrt(1-8*x))/(4*x^2*(1+x)) )); // G. C. Greubel, Nov 23 2022

Rest@CoefficientList[InverseSeries[Series[ -((1+5*n+2*n^2-(1+2*n)*Sqrt[1+6*n+n^2] )/(4*n^2)), {n, 0, 28}]], n] or Length/@Flatten/@NestList[ # /. k_Integer :> Range[ -1-Abs[k], Abs[k]+1]&, {1}, 8]
Flatten[{1,RecurrenceTable[{(n+2)*(7*n-5)*a[n] == (7*n-2)*(7*n-1)*a[n-1] + 4*(2*n-1)*(7*n+2)*a[n-2],a[1]==5,a[2]==27},a,{n,20}]}] (* Vaclav Kotesovec, Oct 14 2012 *)

{a(n) = my(L); L = [1]; if(n < 0, 0, for(i = 1, n, L = concat([ vector(3 + 2*abs(k), i, i - abs(k) - 2) | k <- L])); #L)}; /* Michael Somos, Nov 23 2022 */

def A084076_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x -(1-x)*sqrt(1-8*x))/(4*x^2*(1+x)) ).list()
A084076_list(40) # G. C. Greubel, Nov 23 2022

G.f. is the series reversion of -((1 + 5*x + 2*x^2 - (1 + 2*x)*sqrt(1 + 6*x + x^2))/(4*x^2)).
G.f.: 2*((c(2*x))^3)/(1+c(2*x)) with the o.g.f. c(x) of A000108 (Catalan numbers).
a(n) = Sum_{j=1..n+1} A115195(n, j), n >= 0.
G.f.: (-1 + (1-x)*c(2*x))/(x*(1+x)); cf. A115139. - Wolfdieter Lang, Feb 23 2006
D-finite with recurrence: (n+2)*(7*n-5)*a(n) = (7*n-2)*(7*n-1)*a(n-1) + 4*(2*n-1)*(7*n+2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 7*2^(3n+3)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
D-finite with recurrence (n+2)*a(n) = 2*(4*n+1)*a(n-1) + (n+16)*a(n-2) - 4*(2*n-3)*a(n-3). - R. J. Mathar, Mar 10 2022
a(n) = ( (7*n-1)*(7*n-2)*a(n-1) + 4*(2*n-1)*(7*n+2)*a(n-2) )/((n+2)*(7*n-5)), with a(0) = 1, a(1) = 5. - G. C. Greubel, Nov 23 2022

A116880 Generalized Catalan triangle, called CM(1,2).

Original entry on oeis.org

1, 1, 3, 3, 7, 13, 13, 29, 41, 67, 67, 147, 195, 247, 381, 381, 829, 1069, 1277, 1545, 2307, 2307, 4995, 6339, 7379, 8451, 9975, 14589, 14589, 31485, 39549, 45373, 50733, 56829, 66057, 95235, 95235, 205059, 255747, 290691, 320707, 351187, 388099, 446455, 636925

Offset: 0

Wolfdieter Lang, Mar 24 2006

This triangle generalizes the 'new' Catalan triangle A028364 (which could be called CM(1,1); M stands for author Meeussen).

			Triangle begins:
     1;
     1,    3;
     3,    7,   13;
    13,   29,   41,   67;
    67,  147,  195,  247,  381;
   381,  829, 1069, 1277, 1545, 2307;
  2307, 4995, 6339, 7379, 8451, 9975, 14589;

Nathaniel Johnston, Table of n, a(n) for n = 0..2500
Wolfdieter Lang, First 10 rows.

Column m=0 gives A064062.

Row sums give A116881.

lim:=8: c:=(1-sqrt(1-8*x))/(4*x): g:=(1+2*x*c)/(1+x): gf1:=g*(x*c)^m: for m from 0 to lim do t:=taylor(gf1, x, lim+1): for n from 0 to lim do a[n,m]:=coeff(t, x, n):od:od: gf2:=g*sum(a[s,k]*(2*c)^k,k=0..s): for s from 0 to lim do t:=taylor(gf2, x, lim+1): for n from 0 to lim do b[n,s]:=coeff(t, x, n):od:od: seq(seq(b[n-s,s],s=0..n),n=0..lim); # Nathaniel Johnston, Apr 30 2011

G.f. for columns m >= 0 (without leading zeros): c(2;x)*Sum_{k=0..m} C(1,2;m,k)*(2*c(2*x))^k with c(2;x):=(1+2*x*c(2*x))/(1+x) the g.f. of A064062 and c(x) is the g.f. of A000108 (Catalan). C(1,2;n,m) is the triangle A115193(n,m).

A116866 Generalized Catalan triangle of Riordan type, called C(1,3).

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 25, 25, 7, 1, 190, 190, 55, 10, 1, 1606, 1606, 472, 94, 13, 1, 14506, 14506, 4300, 898, 142, 16, 1, 137089, 137089, 40861, 8785, 1495, 199, 19, 1, 1338790, 1338790, 400567, 87826, 15655

Offset: 0

Wolfdieter Lang, Mar 24 2006

This triangle is the second of a family of generalizations of the Catalan convolution triangle A033184 (which belongs to the Bell subgroup of the Riordan group).
The o.g.f. of the row polynomials P(n,x):=sum(a(n,m)*x^n,m=0..n) is D(x,z)=g(z)/(1 - x*z*c(3*z))= g(z)*(3*z-x*z*(1-3*z*c(3*z)))/(3*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(3*x)) with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064063 (C(3;n) Catalan generalization).
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

			[1];[1,1];[4,4,1];[25,25,7,1];[190,190,55,10,1];...
Production matrix begins:
1, 1
3, 3, 1
9, 9, 3, 1
27, 27, 9, 3, 1
81, 81, 27, 9, 3, 1
243, 243, 81, 27, 9, 3, 1
... _Philippe Deléham_, Sep 22 2014

Wolfdieter Lang, First 10 rows.

Row sums give A116867.
Compare with the row reversed and scaled triangle A116868 (called Y(1, 3)).
Cf. A115193 (similar sequence C(1,2)).

G.f. for column m>=0 is g(x)*(x*c(3*x))^m, with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers).

A115194 A sequence related to A000108 (Catalan numbers).

Original entry on oeis.org

1, 7, 45, 291, 1917, 12867, 87805, 607747, 4257789, 30140419, 215277565, 1549615107, 11230642173, 81882660867, 600196448253, 4420404117507, 32695452696573, 242766809923587, 1808890431799293, 13521381274681347

Offset: 0

Wolfdieter Lang, Feb 23 2006

Also one fourth of third diagonal of triangle A115195, called Y(1,2).

Fourth column (m=3) of triangle A115193, called C(1,2).

L. Guo, W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337, sequence d(n).

G.f.: (-1+x + (1-3*x)*c(2*x))/(2*(1+x)*x^2), with the o.g.f. c(x) of A000108 (Catalan).
a(n)= A115193(n+3,3), n>=0.
D-finite with recurrence -(n+3)*(5*n-2)*a(n) +(35*n^2+31*n+18)*a(n-1) +4*(5*n+3)*(2*n+1)*a(n-2)=0. - R. J. Mathar, Jul 21 2017

A115197 Convolution of generalized Catalan numbers A064062 (called C(n;2)).

Original entry on oeis.org

1, 2, 7, 32, 169, 974, 5947, 37820, 247885, 1662890, 11362399, 78806936, 553386097, 3926523782, 28108587139, 202764451700, 1472446595221, 10755543924578, 78973277044903, 582558618222416, 4315238786662585

Offset: 0

Wolfdieter Lang, Feb 23 2006

nonn

Row sums of triangle A115193, called C(1,2).

The o.g.f. given below follows from the Riordan matrix structure of the triangle A115193. See the o.g.f. for the row polynomials of A115193.

Joseph Abate and Ward Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, after theorem 5.
Owen John Levens, Joel Brewster Lewis, and Bridget Eileen Tenner, Global patterns in signed permutations, arXiv:2504.13108 [math.CO], 2025. See p. 19.

a(n)= sum(A115193(n,m),m=0..n), n>=0.
G.f.: ((1+2*x*c(2*x))/(1+x))^2 = ((1-2*x) + 6*x*c(2*x))/(1+x)^2, with the o.g.f. c(x) of Catalan numbers A000108.
a(n)= sum(C(2;n-k)*C(2;k),k=0..n), n>=0, with C(2;n):= A064062(n).
a(n)=4*A178792(n)-3*(n+1)*A064062(n+1) [From Joseph Abate, Jun 21 2010]
n*a(n) +(-7*n+13)*a(n-1) +4*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

cp's OEIS Frontend

A115202 Fifth column of triangle A115193 (called C(1,2)).

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A115204 Seventh column of triangle A115193 (called C(1,2)).

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A115203 Sixth column of triangle A115193 (called C(1,2)).

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A084076 Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.

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A116880 Generalized Catalan triangle, called CM(1,2).

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A116866 Generalized Catalan triangle of Riordan type, called C(1,3).

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A115194 A sequence related to A000108 (Catalan numbers).

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A115197 Convolution of generalized Catalan numbers A064062 (called C(n;2)).

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