A115195 -id:A115195 - cp's OEIS frontend

A115193 Generalized Catalan triangle of Riordan type, called C(1,2).

1, 1, 1, 3, 3, 1, 13, 13, 5, 1, 67, 67, 27, 7, 1, 381, 381, 157, 45, 9, 1, 2307, 2307, 963, 291, 67, 11, 1, 14589, 14589, 6141, 1917, 477, 93, 13, 1, 95235, 95235, 40323, 12867, 3363, 723, 123, 15, 1, 636925

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle is the first 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_{m=0..n} a(n,m)*x^n is D(x,z) = g(z)/(1 - x*z*c(2*z)) = g(z)*(2*z-x*z*(1-2*z*c(2*z)))/(2*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(2*x)) with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064062 (C(2;n) Catalan generalization).
The column sequences (without leading zeros) are A064062, A064062(n+1), A084076, A115194, A115202-A115204, for m=0..6.
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

			Triangle begins:
   1;
   1,  1;
   3,  3,  1;
  13, 13,  5,  1;
  67, 67, 27,  7,  1;
  ...
Production matrix begins:
    1,   1;
    2,   2,   1;
    4,   4,   2,   1;
    8,   8,   4,   2,   1;
   16,  16,   8,   4,   2,   1;
   32,  32,  16,   8,   4,   2,   1;
   64,  64,  32,  16,   8,   4,   2,   1;
  128, 128,  64,  32,  16,   8,   4,   2,   1;
  ... _Philippe Deléham_, Sep 22 2014

Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (up to row 100)
Wolfdieter Lang, First 10 rows.

Row sums give A115197. Compare with the row reversed and scaled triangle A115195.

Cf. A116866 (similar sequence C(1,3)).

lim:=7: 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: seq(seq(a[n,m],m=0..n),n=0..lim); # Nathaniel Johnston, Apr 30 2011

A110510[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k);
T[n_, k_] := If[n == 0, 1, Sum[A110510[n, i], {i, k, n}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 17 2025 *)

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

T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - Werner Schulte, Mar 24 2019

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

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

Original entry on oeis.org

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

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

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A115193 Generalized Catalan triangle of Riordan type, 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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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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A115194 A sequence related to A000108 (Catalan numbers).

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