A126932 -id:A126932 - cp's OEIS frontend

A126954 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1.

1, 3, 1, 10, 4, 1, 34, 15, 5, 1, 117, 54, 21, 6, 1, 405, 192, 81, 28, 7, 1, 1407, 678, 301, 116, 36, 8, 1, 4899, 2386, 1095, 453, 160, 45, 9, 1, 17083, 8380, 3934, 1708, 658, 214, 55, 10, 1, 59629, 29397, 14022, 6300, 2580, 927, 279, 66, 11, 1

Offset: 0

Philippe Deléham, Mar 19 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

			Triangle begins:
     1;
     3,    1;
    10,    4,    1;
    34,   15,    5,   1;
   117,   54,   21,   6,   1;
   405,  192,   81,  28,   7,  1;
  1407,  678,  301, 116,  36,  8, 1;
  4899, 2386, 1095, 453, 160, 45, 9, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 3, 1], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k=0..n} T(n,k) = A126932(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A059738(m+n).
Sum_{k=0..n} T(n,k)*(-k+1) = 3^n. - Philippe Deléham, Mar 26 2007

A141223 Expansion of 1/(sqrt(1-4x)(1-3xc(x))), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 5, 24, 113, 526, 2430, 11166, 51105, 233190, 1061510, 4822984, 21879786, 99135076, 448707992, 2029215114, 9170247393, 41416383366, 186957126702, 843575853984, 3804927658878, 17156636097156, 77339426905812, 348553445817084, 1570548863858778, 7075531788285276

Offset: 0

Paul Barry, Jun 14 2008

Binomial transform of A126932. Hankel transform is (-1)^n.

Row sums of the Riordan matrix (1/(1-4*x),(1-sqrt(1-4*x))/(2*sqrt(1-4*x))) (A188481). - Emanuele Munarini, Apr 01 2001

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A000108.
Cf. A000302, A000984, A026641.
Cf. A006256, A385605, A385632.

CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* Emanuele Munarini, Apr 01 2011 *)

makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* Emanuele Munarini, Apr 01 2011 */

a(n) = Sum_{k=0..n} C(2*n-k,n-k)*3^k.
From Emanuele Munarini, Apr 01 2011: (Start)
a(n) = [x^n] 1/((1-x)^(n+1) * (1-3*x)). [Corrected by Seiichi Manyama, Aug 03 2025]
a(n) = 3^(2*n+1)/2^(n+2) + (1/4)*Sum_{k=0..n} binomial(2*k,k)*(9/2)^(n-k).
D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.
G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)
G.f.: (1/(1-4*x)^(1/2)+3)/(4-18*x) = (2 + x/(Q(0)-2*x))/(2-9*x) where Q(k) = 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 18 2013
a(n) ~ 3^(2*n + 1) / 2^(n + 1). - Vaclav Kotesovec, Sep 15 2021
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k). - Seiichi Manyama, Aug 03 2025
a(n) = 3^(2*n+1)*2^(-n-1) - binomial(2*n+1, n)*(hypergeom([1, -1-n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k). - Seiichi Manyama, Aug 07 2025

A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 19, 9, 5, 1, 1, 51, 26, 11, 6, 1, 1, 141, 70, 34, 13, 7, 1, 1, 393, 197, 92, 43, 15, 8, 1, 1, 1107, 553, 265, 117, 53, 17, 9, 1, 1, 3139, 1570, 751, 346, 145, 64, 19, 10, 1, 1, 8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 26 2009

Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A002426 and where g(x) is the g.f. of A005043. - Philippe Deléham, Dec 05 2009

Matrix product P * Q * P^(-1), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158815 and A171243. - Peter Bala, Jul 13 2021

			First rows of the triangle:
     1;
     1,    1;
     3,    1,    1;
     7,    4,    1,   1;
    19,    9,    5,   1,   1;
    51,   26,   11,   6,   1,   1;
   141,   70,   34,  13,   7,   1,  1;
   393,  197,   92,  43,  15,   8,  1,  1;
  1107,  553,  265, 117,  53,  17,  9,  1,  1;
  3139, 1570,  751, 346, 145,  64, 19, 10,  1, 1;
  8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1;

T(n, 0) = A002426(n), A005773 (row sums).

Cf. A046899, A007318, A158815, A171243.

A158793 := proc (n, k)
  add((-1)^(n+j)*binomial(n, j)*binomial(2*j-k, j-k), j = k..n);
end proc:
seq(seq(A158793(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T[n_, k_] := (-1)^(k + n) Binomial[n, k] HypergeometricPFQ[{k/2 + 1/2, k/2 + 1, k - n}, {k + 1, k + 1}, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Jul 17 2021 *)

T(n, m) = Sum_{k=m..n-1} A130595(n,k) * A092392(k+1,m+1), with the triangular interpretation of A092392.
Conjecture: T(n,1) = A113682(n-1). - R. J. Mathar, Oct 06 2009
Sum_{k=0..n} T(n,k)*x^k = A002426(n), A005773(n+1), A000244(n), A126932(n) for x = 0,1,2,3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = (-1)^(k + n) binomial(n, k) hypergeom([k/2 + 1/2, k/2 + 1, k - n], [k + 1, k + 1], 4). - Peter Luschny, Jul 17 2021

Simplified definition from R. J. Mathar, Oct 06 2009

A301477 T(n,k) = Sum_{j=0..n-k} H(n,j)2^k with H(n,k) = binomial(n,k) hypergeom([-k/2, 1/2-k/2], [2-k+n], 4), for 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 2, 2, 5, 6, 4, 13, 18, 16, 8, 35, 52, 56, 40, 16, 96, 150, 180, 160, 96, 32, 267, 432, 560, 568, 432, 224, 64, 750, 1246, 1708, 1904, 1680, 1120, 512, 128, 2123, 3600, 5152, 6160, 6048, 4736, 2816, 1152, 256, 6046, 10422, 15432, 19488, 20736, 18240, 12864, 6912, 2560, 512

Offset: 0

Peter Luschny, Mar 22 2018

			                               1
                              2, 2
                            5, 6, 4
                         13, 18, 16, 8
                       35, 52, 56, 40, 16
                   96, 150, 180, 160, 96, 32
                267, 432, 560, 568, 432, 224, 64
          750, 1246, 1708, 1904, 1680, 1120, 512, 128
      2123, 3600, 5152, 6160, 6048, 4736, 2816, 1152, 256

Row sums are A126932, first column A005773, diagonal A000079.

Cf. A301475 (general case).

H := (n,k) -> binomial(n,k)*hypergeom([-k/2,1/2-k/2],[2-k+n], 4):
T := (n,k) -> add(simplify(H(n,j)*2^k), j=0..n-k):
seq(seq(T(n,k), k=0..n), n=0..9);

s={};For[n=0,n<13,n++,For[k=0,kDetlef Meya, Oct 03 2023 *)

cp's OEIS Frontend

A126954 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1.

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A141223 Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.

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A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.

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A301477 T(n,k) = Sum_{j=0..n-k} H(n,j)*2^k with H(n,k) = binomial(n,k)* hypergeom([-k/2, 1/2-k/2], [2-k+n], 4), for 0 <= k <= n, triangle read by rows.

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Maple

Mathematica

A141223 Expansion of 1/(sqrt(1-4x)(1-3xc(x))), where c(x) is the g.f. of A000108.

A301477 T(n,k) = Sum_{j=0..n-k} H(n,j)2^k with H(n,k) = binomial(n,k) hypergeom([-k/2, 1/2-k/2], [2-k+n], 4), for 0 <= k <= n, triangle read by rows.