A094531 -id:A094531 - cp's OEIS frontend

A094527 Triangle T(n,k), read by rows, defined by T(n,k) = binomial(2*n,n-k).

1, 2, 1, 6, 4, 1, 20, 15, 6, 1, 70, 56, 28, 8, 1, 252, 210, 120, 45, 10, 1, 924, 792, 495, 220, 66, 12, 1, 3432, 3003, 2002, 1001, 364, 91, 14, 1, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1, 184756, 167960

Offset: 0

Paul Barry, May 07 2004

Right-hand side of even-numbered rows of Pascal's triangle.
Row sums are A032443. Reverse of A062344. Right-hand side of A034870. Binomial transform of trinomial triangle A094531.
Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + 2*T(n-1,1), T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 14 2007
Central coefficients T(2n,n) are binomial(4n,n) (A005810).
The A- and Z-sequence for this Riordan triangle is [1,2,1] and [2,2], respectively. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. See also the Philippe Deléham comment above. - Wolfdieter Lang, Nov 22 2012

			The triangle T(n,k) begins:
  n\k      0      1      2     3     4     5    6    7   8  9 10
  0:       1
  1:       2      1
  2:       6      4      1
  3:      20     15      6     1
  4:      70     56     28     8     1
  5:     252    210    120    45    10     1
  6:     924    792    495   220    66    12    1
  7:    3432   3003   2002  1001   364    91   14    1
  8:   12870  11440   8008  4368  1820   560  120   16   1
  9:   48620  43758  31824 18564  8568  3060  816  153  18  1
  10: 184756 167960 125970 77520 38760 15504 4845 1140 190 20  1
  ... Reformatted ad extended by _Wolfdieter Lang_, Nov 22 2012
From _Paul Barry_, Sep 07 2009: (Start)
Production array is
  2, 1,
  2, 2, 1,
  0, 1, 2, 1,
  0, 0, 1, 2, 1,
  0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 0, 1, 2, 1 (End)
From _Wolfdieter Lang_, Nov 22 2012: (Start)
Recurrence from the Riordan A-sequence [1,2,1]: T(4,1) = 56 = 1*T(3,0) + 2*T(3,1) + 1*T(3,2) = 1*20 + 2*15 + 1*6.
Recurrence from the Riordan Z-sequence [2,2]: T(7,0) = 3432 = 2*T(6,0) + 2*T(6,1) = 2*924 + 2*792. See the _Philippe Deléham_ comment above. (End)

Indranil Ghosh, Rows 0..100 of triangle, flattened
Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 19.
A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012. - From _N. J. A. Sloane_, Sep 16 2012
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263
T. M. Richardson, The Reciprocal Pascal Matrix, arXiv preprint arXiv:1405.6315 [math.CO], 2014.

Cf. A007318, A032443, A039599, A039598.

A094527 := proc(n,k)
    binomial(2*n,n-k) ;
end proc: # R. J. Mathar, Jun 04 2013

T[n_, k_] := Binomial[2*n, n - k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017 *)

Riordan array (1/sqrt(1-4*x), (1-2*x-sqrt(1-4*x))/(2*x)). Column k has e.g.f. exp(2*x)*Bessel_I(k, 2*x). - Paul Barry, Jul 14 2005
Product of Riordan arrays (1/(1-x), x/(1-x)) (Pascal's triangle, A007318) and (1/sqrt(1-2x-3x^2), (1-x-sqrt(1-2x-3x^2))/(2x)) (A094531). Inverse is A110162. - Paul Barry, Jul 14 2005
T(n,k) = Sum_{j=0..n} C(n,j)*C(n,j-k). - Paul Barry, Mar 07 2006
T(n,k) = Sum_{h>=k} A039599(n,h). Sum_{k=0..n} T(n,k) = A032443(n). - Philippe Deléham, May 01 2006
Sum_{k=0..n} T(n,k)^2 = A036910(n). - Philippe Deléham, May 07 2006
Sum_{k=0..n} T(n,k)*(-1)^k = A088218(n). - Philippe Deléham, Mar 14 2007
From Wolfdieter Lang, Nov 22 2012: (Start)
The o.g.f. for the row polynomials P(n,x) := Sum_{k=0..n} T(n,k)*x^k is G(z,x) = (-x + (1+x)*z + x*z*c(z))/(sqrt(1-4*z)*((1+x)^2*z -x)) with c the o.g.f. of A000108 (Catalan). This follows from the Riordan property.
The o.g.f. for column no. k is (c(x)-1)^k/sqrt(1-4*x) (from the Riordan property). (End)
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = ( 1 - 2*x - sqrt(1 - 4*x) )/(2*x) and so belongs to the hitting time subgroup of the Riordan group (see Peart and Woan, Example 5.1).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = (1 + x)^2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
From Peter Bala, Jul 21 2015: (Start)
n-th row polynomial R(n,t) = [x^n] ( (1 + (1 + t)*x)^2/(1 + t*x) )^n.
exp ( Sum_{n >= 1} R(n,t)*x^n/n ) = 1 + (2 + t)*x + (5 + 4*t + t^2)*x^2 + ... is the o.g.f. for A039598. (End)

Entry revised by N. J. A. Sloane, Mar 23 2007

A113682 Expansion of 2/(sqrt(1-2x-3x^2)(1+x+sqrt(1-2x-3*x^2))).

Original entry on oeis.org

1, 1, 4, 9, 26, 70, 197, 553, 1570, 4476, 12827, 36894, 106471, 308113, 893804, 2598313, 7567466, 22076404, 64498427, 188689684, 552675365, 1620567763, 4756614062, 13974168190, 41088418151, 120906613075, 356035078102

Offset: 0

Paul Barry, Nov 04 2005

Convolution of A002426 and A005043. Diagonal sums of A094531.
Hankel transform is A164611. - Paul Barry, Aug 17 2009
David Scambler observed that [1,0,a(n-2)] for n>=2 count the Dyck paths of semilength n such that the number of peaks equals the number of hills plus the number of returns. - Peter Luschny, Oct 22 2012
Conjectural congruences (working with an offset of 1): a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. - Peter Bala, Mar 15 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Cf. A002426, A005043, A094531, A217539, A217540, A246437, A005717.

[(Evaluate(GegenbauerPolynomial(n+1, -n-1), -1/2) + (-1)^n)/2: n in [0..40]];  // G. C. Greubel, Apr 04 2024

ex[x_]:=Module[{sx=Sqrt[1-2x-3x^2]},2/(sx (1+x+sx))]; CoefficientList[ Series[ ex[x],{x,0,40}],x] (* Harvey P. Dale, May 28 2012 *)
Flatten[{1, Table[Coefficient[Sum[(1 + x + x^2)^k, {k, 0, n}], x^n], {n, 1, 30}]}] (* Vaclav Kotesovec, Jan 08 2016 *)

makelist((ultraspherical(n+1,-n-1,-1/2)+(-1)^n)/2,n,0,12); /* Emanuele Munarini, Dec 20 2016 */

x='x+O('x^50); Vec(2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2)))) \\ G. C. Greubel, Feb 28 2017

[(gegenbauer(n+1,-n-1,-1/2) +(-1)^n)/2 for n in range(41)] # G. C. Greubel, Apr 04 2024

a(n) = Sum_{k=0..floor(n/2)} ( Sum_{i=0..n-k} C(n-2k-i, i)*C(n-k, k+i) ).
a(n) = Sum_{k=0..n} A002426(k)*A005043(n-k).
a(n) = Sum_{k=0..n} C(n+1,k+1)*C(k,n-k). - Paul Barry, Aug 21 2007
a(n) = (A002426(n+1) + (-1)^n)/2. - Paul Barry, Aug 17 2009
G.f.: d/dx log(1/(1-x*A005043(x))). - Vladimir Kruchinin, Apr 18 2011
D-finite with recurrence: (n+1)*a(n) +(-n-1)*a(n-1) +(-5*n+1)*a(n-2) +3*(-n+1)*a(n-3)=0. - R. J. Mathar, Nov 26 2012
Recurrence: (n+4)*a(n+3)-(n+4)*a(n+2)-(5*n+14)*a(n+1)-3*(n+2)*a(n)=0. Remark: this recurrence can be obtained using the identity a(n) = (t(n+1)+(-1)^n)/2 and the recurrence of the central trinomial coefficients t(n) = A002426(n). So, the above P-finite recurrences are true. - Emanuele Munarini, Dec 20 2016
a(n) = (-1)^(n+1) * (hypergeom([1/2, -n-1], [1], 4) - 1)/2. - Vladimir Reshetnikov, Apr 25 2016
a(n) = (-1)^n + A246437(n+1). - Vladimir Reshetnikov, Apr 25 2016

A102587 T(n, k) = (-1)^n2[x^k] ChebyshevT(n, (1 - x)/2) with T(0,0) = 1, for 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 2, 0, -3, 1, -1, 4, 2, -4, 1, -1, -5, 5, 5, -5, 1, 2, 0, -12, 4, 9, -6, 1, -1, 7, 7, -21, 0, 14, -7, 1, -1, -8, 12, 24, -30, -8, 20, -8, 1, 2, 0, -27, 9, 54, -36, -21, 27, -9, 1, -1, 10, 15, -60, -15, 98, -35, -40, 35, -10, 1, -1, -11, 22, 66, -99, -77, 154, -22, -66, 44, -11, 1, 2, 0, -48, 16, 180, -120, -196, 216, 9

Offset: 0

Paul D. Hanna, Jan 22 2005

Previous name: Triangular matrix, read by rows, equal to the matrix inverse of triangle A094531, which is the right-hand side of trinomial table A027907.
Riordan array ((1-x^2)/(1+x+x^2),x/(1+x+x^2)). - Paul Barry, Jul 14 2005
Inverse of A094531. Rows sums are 1,0,-2,0,2,0,-2,... with g.f. (1-x^2)/(1+x^2). Diagonal sums are (-1)^n*C(1,n) with g.f. 1-x. - Paul Barry, Jul 14 2005
Row sums form the period 4 sequence: {1, 0,-2,0,2, 0,-2,0,2, ...}. Absolute row sums form A102588.
Sum_{k=0..n} T(n,k)^2 = 2*A002426(n) for n>0.

			Rows begin:
  [1],
  [ -1,1],
  [ -1,-2,1],
  [2,0,-3,1],
  [ -1,4,2,-4,1],
  [ -1,-5,5,5,-5,1],
  [2,0,-12,4,9,-6,1],
  [ -1,7,7,-21,0,14,-7,1],
  [ -1,-8,12,24,-30,-8,20,-8,1],
  [2,0,-27,9,54,-36,-21,27,-9,1],
  [ -1,10,15,-60,-15,98,-35,-40,35,-10,1],
  [ -1,-11,22,66,-99,-77,154,-22,-66,44,-11,1],
  ...

Feihu Liu, Ying Wang, Yingrui Zhang, and Zihao Zhang,Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results, arXiv:2503.17187 [math.CO], 2025. See p. 10.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A094531 (matrix inverse), A102588, A002426.

Table[If[n==0, 1, CoefficientList[(-1)^n 2 ChebyshevT[n, (1-x)/2], x]], {n, 0, 9}] // Flatten (* Peter Luschny, Mar 07 2018 *)

{T(n,k)=local(A); A=matrix(n+1,n+1,r,c,if(r

tabl(nn) = {my(m = matrix(nn, nn, n, k, n--; k--; sum(j=0, n, binomial(n,j)*binomial(j,n-k-j)))^(-1)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Jun 30 2015

T(n,k) = T(n-1,k-1) - T(n-1,k) - T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = -1, T(2,1) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 22 2014
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x + x^2) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = ( 1 - x + sqrt(1 - 2*x - 3*x^2) )/2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

New name by Peter Luschny, Mar 07 2018

A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 17, 20, 13, 5, 1, 1, 38, 50, 38, 19, 6, 1, 1, 89, 126, 107, 63, 26, 7, 1, 1, 216, 322, 296, 196, 96, 34, 8, 1, 1, 539, 834, 814, 588, 326, 138, 43, 9, 1, 1, 1374, 2187, 2236, 1728, 1052, 507, 190, 53, 10, 1

Offset: 0

Philippe Deléham, Nov 07 2009

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  8,  8,  4,  1;
  1, 17, 20, 13,  5, 1;
  1, 38, 50, 38, 19, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Antidiagonal sums give A082395.
Row sums give A383527.
Diagonals include: A006416, A034856, A086615, A140662.
Cf. A001006, A003462, A064189, A082397, A094531.

T:= proc(n, k) option remember; `if`(k=0, 1,
      `if`(k>n, 0, T(n-1, k-1)+T(n-1, k)+T(n-1, k+1)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Apr 20 2018

T[, 0] = T[n, n_] = 1;
T[n_, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019 *)

T(n,0)=1, T(0,k)=0 for k>0, T(n,k)=0 if k>n, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1).
Sum_{k=0..n} k * T(n,k) = A003462(n). - Alois P. Heinz, Apr 20 2018
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = A082397(n-2) for n>=2. - Alois P. Heinz, May 02 2025

A111960 Renewal array for central trinomial numbers A002426.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 7, 3, 1, 19, 20, 12, 4, 1, 51, 61, 40, 18, 5, 1, 141, 182, 135, 68, 25, 6, 1, 393, 547, 441, 251, 105, 33, 7, 1, 1107, 1640, 1428, 888, 420, 152, 42, 8, 1, 3139, 4921, 4572, 3076, 1596, 654, 210, 52, 9, 1, 8953, 14762, 14535, 10456, 5880, 2652, 966, 280, 63, 10, 1

Offset: 0

Paul Barry, Aug 23 2005

Also the convolution triangle of A002426. - Peter Luschny, Oct 06 2022

			Triangle T(n,k) begins:
   1;
   1,  1;
   3,  2,  1;
   7,  7,  3,  1;
  19, 20, 12,  4, 1;
  51, 61, 40, 18, 5, 1;
  ...
From _Paul Barry_, May 12 2009: (Start)
Production matrix is
  1, 1,
  2, 1, 1,
  0, 2, 1, 1,
  -2, 0, 2, 1, 1,
  0, -2, 0, 2, 1, 1,
  4, 0, -2, 0, 2, 1, 1. (End)

Row sums are A111961.
Diagonal sums are A111962.
Inverse is A111963.
Factors as A007318*A111959.
Column k=0 gives A002426.
Cf. A026325.

# Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> A002426(n - 1)); # Peter Luschny, Oct 06 2022

Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).
From Paul Barry, May 12 2009: (Start)
Equals ((1-x^2)/(1+x+x^2),x/(1+x+x^2))^{-1}*(1,x/(1-x^2))=A094531*(1,x/(1-x^2)).
Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));
T(n,k) = Sum_{j=0..n} C(n,j)*C((j-1)/2,(j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
G.f.: 1/(1-xy-x-2x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). (End)

A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} binomial(n+3j,3j)x^j, in row n, column k.

Original entry on oeis.org

1, 1, 2, 1, 7, 1, 1, 16, 10, 1, 30, 45, 5, 1, 50, 141, 50, 1, 1, 77, 357, 266, 28, 1, 112, 784, 1016, 266, 8, 1, 156, 1554, 3139, 1554, 156, 1, 1, 210, 2850, 8350, 6765, 1452, 55, 1, 275, 4917, 19855, 24068, 9042, 880, 11

Offset: 0

Roger L. Bagula, May 30 2010

Every third row is symmetrical.
Row sums are 3^n.
2*k instead of 3*k in the binomial() gives A034839 with alternating rows of A086645.

			1;
1, 2;
1, 7, 1;
1, 16, 10;
1, 30, 45, 5;
1, 50, 141, 50, 1;
1, 77, 357, 266, 28;
1, 112, 784, 1016, 266, 8;
1, 156, 1554, 3139, 1554, 156, 1;
1, 210, 2850, 8350, 6765, 1452, 55;
1, 275, 4917, 19855, 24068, 9042, 880, 11;

Cf. A094531, A111808, A027907.

A178618 := proc(n,k)
    (1-x)^(n+1)*add( binomial(n+3*j,3*j)*x^j,j=0..n+1) ;
    coeftayl(%,x=0,k) ;
end proc:
seq(seq(A178618(n,k),k=0..n),n=0..8) ; # R. J. Mathar, Nov 05 2012

p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 3*k, 3*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]

A160905 Right hand side of Pascal rhombus A059317.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 9, 8, 3, 1, 29, 22, 13, 4, 1, 82, 72, 42, 19, 5, 1, 255, 218, 146, 70, 26, 6, 1, 773, 691, 476, 261, 107, 34, 7, 1, 2410, 2158, 1574, 914, 428, 154, 43, 8, 1, 7499, 6833, 5122, 3177, 1603, 659, 212, 53, 9, 1, 23575, 21612, 16706, 10816, 5867, 2628, 967

Offset: 0

Paul Barry, May 29 2009

Riordan array (1/sqrt((1+x-x^2)*(1-3*x-x^2)), (1-x-x^2-sqrt((1+x-x^2)*(1-3*x-x^2)))/(2*x)). Can be factored as
(1/(1-x-x^2), x/(1-x-x^2))*(1/sqrt(1-4x^2),xc(x^2)) = (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x))*(1/sqrt(1-4x^2),xc(x^2))
and (1/(1-x^2),x/(1-x^2))*(1/sqrt(1-2x-3x^2),(1-x-sqrt(1-2x-3x^2))/(2x)).
Here, c(x) is the g.f. of the Catalan numbers A000108.
A160905 = A037027*A108044 = abs(A049310)*A007318*A108044 = abs(A049310)*A094531.

			Triangle begins:
    1;
    1,   1;
    4,   2,   1;
    9,   8,   3,  1;
   29,  22,  13,  4,  1;
   82,  72,  42, 19,  5, 1;
  255, 218, 146, 70, 26, 6, 1;
  ...

Left column gives A059345.

Cf. A059317.

Number triangle T(n,k) = Sum_{i=0..n} (Sum_{j=0..n} C((n+j)/2,j)*C(j,i)*(1+(-1)^(n-j))/2)*C(i,(i-k)/2)*(1+(-1)^(i-k))/2;

T(n,k) = Sum_{j=0..n} C((n+j)/2,j)*((1+(-1)^(n-j))/2)*Sum_{i=0..j} C(j,i)*C(i,j-k-i).

cp's OEIS Frontend

A102588 Absolute row sums of triangle A102587, which is equal to the matrix inverse of triangle A094531 (the right-hand side of trinomial table A027907).

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A094527 Triangle T(n,k), read by rows, defined by T(n,k) = binomial(2*n,n-k).

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A113682 Expansion of 2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2))).

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A102587 T(n, k) = (-1)^n*2*[x^k] ChebyshevT(n, (1 - x)/2) with T(0,0) = 1, for 0 <= k <= n, triangle read by rows.

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A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.

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A111960 Renewal array for central trinomial numbers A002426.

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A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} *binomial(n+3*j,3*j)*x^j, in row n, column k.

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Mathematica

A113682 Expansion of 2/(sqrt(1-2x-3x^2)(1+x+sqrt(1-2x-3*x^2))).

A102587 T(n, k) = (-1)^n2[x^k] ChebyshevT(n, (1 - x)/2) with T(0,0) = 1, for 0 <= k <= n, triangle read by rows.

A178618 Triangle T(n,k) with the coefficient [x^k] of the series (1-x)^(n+1) * sum_{j=0..infinity} binomial(n+3j,3j)x^j, in row n, column k.