A321620 -id:A321620 - cp's OEIS frontend

A116392 Riordan array (1/sqrt(1-2x-3x^2), 1/sqrt(1-2x-3x^2) -1).

1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 19, 42, 32, 10, 1, 51, 131, 128, 60, 13, 1, 141, 406, 475, 292, 97, 16, 1, 393, 1247, 1685, 1267, 561, 143, 19, 1, 1107, 3814, 5800, 5112, 2804, 962, 198, 22, 1, 3139, 11623, 19540, 19624, 12748, 5464, 1522, 262, 25, 1, 8953, 35334

Offset: 0

Paul Barry, Feb 12 2006

Triangle, read by rows, given by [1, 2, -1, -1, 2, 1/2, 1/2, 2, -1, -1, 2, 1/2, 1/2, 2, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 11 2020

			Triangle begins:
   1;
   1,   1;
   3,   4,   1;
   7,  13,   7,  1;
  19,  42,  32, 10,  1;
  51, 131, 128, 60, 13, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Row sums are A115967. Diagonal sums are A116394.

Cf. A321620.

[[(&+[ Binomial(n,m)*(&+[ (&+[ Round((-1)^(k-j)*4^r* Binomial(k,j)*Binomial(j, m-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]): m in [0..n]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 23 2019

# The function RiordanSquare is defined in A321620.
RiordanSquare(1/sqrt(1 - 2*x - 3*x^2), 10); # Peter Luschny, Feb 15 2020

t[n_,k_]:= Sum[(-1)^(k-j)*Binomial[k,j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r,0,Floor[n/2]}], {j,0,k}]; Table[Sum[Binomial[n, j]*t[j,k], {j,0,n}] {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 23 2019 *)

t(n,k) = sum(j=0,k, sum(r=0,floor(n/2), (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) ));
T(n,k) = sum(j=0,n, binomial(n,j)*t(j,k)); \\ G. C. Greubel, May 23 2019

[[sum(binomial(n,m)*sum( sum( (-1)^(k-j)*4^r* binomial(k,j)* binomial(r+(j-1)/2, r)*binomial(j, m-2*r) for r in (0..floor(n/2))) for j in (0..k)) for m in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 23 2019

Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*A116389(j,k).

A132372 T(n, k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k. Triangle T(n,k), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 5, 1, 22, 38, 22, 7, 1, 90, 158, 98, 38, 9, 1, 394, 698, 450, 194, 58, 11, 1, 1806, 3218, 2126, 978, 334, 82, 13, 1, 8558, 15310, 10286, 4942, 1838, 526, 110, 15, 1, 41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1

Offset: 0

Philippe Deléham, Nov 20 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .
Transpose of triangular array A033878. - Michel Marcus, May 02 2015
The triangle is the Riordan square (A321620) of A155069. - Peter Luschny, Feb 01 2020

			Triangle begins:
      1;
      1,     1;
      2,     3,     1;
      6,    10,     5,     1;
     22,    38,    22,     7,    1;
     90,   158,    98,    38,    9,    1;
    394,   698,   450,   194,   58,   11,   1;
   1806,  3218,  2126,   978,  334,   82,  13,   1;
   8558, 15310, 10286,  4942, 1838,  526, 110,  15,  1;
  41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1 ; ...
...
The production matrix M begins:
  1, 1
  1, 2, 1
  1, 2, 2, 1
  1, 2, 2, 2, 1
  1, 2, 2, 2, 2, 1
  ...

Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, Journal of Integer Sequences, Vol. 1 (1998), #98.1.7.

Cf. A006318, A103136 (signed version), A033878 (transpose).

Cf. A155069, A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare((3-x-sqrt(1-6*x+x^2))/2, 10); # Peter Luschny, Feb 01 2020
# Alternative:
A132372 := proc(dim) # dim is the number of rows requested.
local T, j, A, k, C, m; m := 1;
T := [seq([seq(0, j = 0..k)], k = 0..dim-1)];
A := [seq(ifelse(k = 0, 1 + x, 2 - irem(k, 2)), k = 0..dim-2)];
C := [seq(1, k = 1..dim+1)]; C[1] := 0;
for k from 0 to dim - 1 do
    for j from k + 1 by -1 to 2 do
        C[j] := C[j-1] + C[j+1] * A[j-1] od;
    T[m] := [seq(coeff(C[2], x, j), j = 0..k)];
    m := m + 1
od; ListTools:-Flatten(T) end:
A132372(10);  # Peter Luschny, Nov 16 2023

Sum_{k, 0<=k<=n} T(n,k) = A006318(n) .

T(n,0) = A155069(n). - Philippe Deléham, Nov 03 2009

A109956 Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -12, 18, -7, 1, 55, -88, 42, -10, 1, -273, 455, -245, 75, -13, 1, 1428, -2448, 1428, -510, 117, -16, 1, -7752, 13566, -8379, 3325, -910, 168, -19, 1, 43263, -76912, 49588, -21252, 6578, -1472, 228, -22, 1, -246675, 444015, -296010, 134550, -45630, 11700, -2223, 297, -25, 1

Offset: 0

Paul Barry, Jul 06 2005

Riordan array (g,f) where f/(1-f)^3=x and g=1-f.
First column is (-1)^n*binomial(3n,n)/(2n+1), a signed version of A001764.
Second column is a signed version of A006629.
Diagonal sums are A109957.

			Triangle begins:
      1;
     -1,   1;
      3,  -4,    1;
    -12,  18,   -7,   1;
     55, -88,   42, -10,   1;
   -273, 455, -245,  75, -13, 1;
   ...

Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 13.
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8 (up to signs).

Cf. A001764, A109955, A109957, A321620.

# Function RiordanSquare defined in A321620.
tt := sin(arcsin(3*sqrt(x*3/4))/3)/sqrt(x*3/4): R := RiordanSquare(tt, 11):
seq(seq(LinearAlgebra:-Row(R,n)[k]*(-1)^(n+k), k=1..n), n=1..11); # Peter Luschny, Nov 27 2018

T[n_, k_] := (-1)^(n - k)((3k + 1)/(2n + k + 1)) Binomial[3n, n - k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

tabl(nn) = {my(m = matrix(nn, nn, n, k, if (nMichel Marcus, Nov 20 2015

Number triangle T(n, k) = (-1)^(n-k)*((3k+1)/(2n+k+1))*binomial(3n, n-k).
From Werner Schulte, Oct 27 2015: (Start)
If u(m,n) = (-1)^n*(Sum_{k=0..n} T(n,k)*((m+1)*k+1)) and v(m,n) = (-1)^n*(Sum_{k=0..n} (-1)^k*T(n,k)*m^k) and D(x) is the g.f. of A001764 then P(m,x) = Sum_{n>=0} u(m,n)*x^n = 1-(m+1)*x*D(x)^2 and Q(m,x) = Sum_{n>=0} v(m,n)*x^n = 1/P(m,x).
If G(k,x) is the g.f. of column k (k>=0) then G(k,x) = G(0,x)^(3*k+1). (End)

A154380 The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 15, 29, 20, 7, 1, 52, 102, 77, 35, 9, 1, 203, 392, 302, 157, 54, 11, 1, 877, 1641, 1235, 683, 277, 77, 13, 1, 4140, 7451, 5324, 2987, 1329, 445, 104, 15, 1, 21147, 36525, 24329, 13391, 6230, 2340, 669, 135, 17, 1

Offset: 0

Paul Barry, Jan 08 2009

The Riordan square is defined in A321620.
Previous name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1, 1, 1, 2, 1, 3, 1, 4, 1, ...] DELTA [1, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
In general, the triangle [r_0, r_1, r_2, ...] DELTA [s_0, s_1, s_2, ...] has generating function
1/(1 - (r_0*x + s_0*x*y)/(1 - (r_1*x + s_1*x*y)/(1 - (r_2*x + s_2*x*y)/(1 -... (continued fraction)
A130167*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 11 2009

			Triangle begins
     1;
     1,   1;
     2,   3,   1;
     5,   9,   5,   1;
    15,  29,  20,   7,  1;
    52, 102,  77,  35,  9,  1;
   203, 392, 302, 157, 54, 11, 1;

Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.

First column are the Bell numbers A000110.
Row sums are A154381, alternating row sums are A000007.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(add(x^k/mul(1-j*x, j=1..k), k=0..10), 10); # Peter Luschny, Dec 06 2018

RiordanSquare[gf_, len_] := Module[{T}, T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[gf, {x, 0, n}], Sum[T[j, k - 1] T[n - j, 0], {j, k - 1, n - 1}]]; Table[T[n, k], {n, 0, len - 1}, {k, 0, n}]];
RiordanSquare[Sum[x^k/Product[1 - j x, {j, 1, k}], {k, 0, 10}], 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

G.f.: 1/(1-(x+xy)/(1-x/(1-x/(1-2x/(1-x/(1-3x/(1-x/(1-4x/(1-... (continued fraction).

New name by Peter Luschny, Dec 06 2018

A172094 The Riordan square of the little Schröder numbers A001003.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 17, 7, 1, 45, 76, 40, 10, 1, 197, 353, 216, 72, 13, 1, 903, 1688, 1145, 458, 113, 16, 1, 4279, 8257, 6039, 2745, 829, 163, 19, 1, 20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1, 103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1

Offset: 0

Philippe Deléham, Jan 25 2010

The Riordan square is defined in A321620.
Previous name was: Triangle, read by rows, given by [1,2,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Riordan array (f(x), f(x)-1) where f(x) is the g.f. of A001003. Equals A122538*A007318.

			Triangle begins:
     1
     1,      1
     3,      4,      1
    11,     17,      7,     1
    45,     76,     40,    10,     1
   197,    353,    216,    72,    13,     1
   903,   1688,   1345,   458,   113,    16,    1
  4279,   8257,   6039,  2745,   829,   163,   19,   1
20793,  41128,  31864, 15932,  5558,  1356,  222,  22,  1
103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
.
Production matrix begins:
1, 1
2, 3, 1
0, 2, 3, 1
0, 0, 2, 3, 1
0, 0, 0, 2, 3, 1
0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 0, 2, 3, 1
... - _Philippe Deléham_, Sep 24 2014

P. Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and J. Int. Seq. 17 (2014) # 14.5.1.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 12.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

T(n, 0) = A001003(n) (little Schröder), A109980 (row sums).
Diagonals: A239204, A000012, A016777.
Cf. A122538, A007318, A321620.

T := (n, k) -> local j; add((binomial(n-1, j)*binomial(n+1, k+j+1) - binomial(n, j)*binomial(n, k+j+1))*2^j, j = 0..n-k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Jan 24 2025

DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x r[[k+1]] + y s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k] p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
nmax = 9;
DELTA[Table[{1, 2}, (nmax+1)/2] // Flatten, Prepend[Table[0, {nmax}], 1], nmax] // Flatten (* Jean-François Alcover, Aug 07 2018 *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[(1 + x - Sqrt[1 - 6x + x^2])/(4x), 11] // Flatten  (* Peter Luschny, Nov 27 2018 *)

T(0, 0) = 1, T(n, k) = 0 if k>n, T(n, 0) = T(n-1, 0) + 2*T(n-1, 1), T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k+1) for k>0.
Sum_{0<=k<=n} T(n, k) = A109980(n).
Sum_{k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n).
T(n, k) = Sum_{j=0..n-k} (binomial(n-1, j)*binomial(n+1, k+j+1) - binomial(n, j)*binomial(n, k+j+1))*2^j. (Cigler) - Peter Luschny, Jan 24 2025

New name by Peter Luschny, Nov 27 2018

A208659 Triangle of coefficients of polynomials v(n,x) jointly generated with A185045; see the Formula section.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 2, 10, 16, 8, 2, 14, 36, 40, 16, 2, 18, 64, 112, 96, 32, 2, 22, 100, 240, 320, 224, 64, 2, 26, 144, 440, 800, 864, 512, 128, 2, 30, 196, 728, 1680, 2464, 2240, 1152, 256, 2, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512, 2, 38, 324

Offset: 1

Clark Kimberling, Mar 03 2012

Alternating row sums:  1, 0, 0, 0, 0, 0, 0, 0, 0, ...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

			First five rows:
  1;
  2,  2;
  2,  6,  4;
  2, 10, 16,  8;
  2, 14, 36, 40, 16;
First five polynomials v(n,x):
  1
  2 +  2x = 2*(1+x)
  2 +  6x +  4x^2 = 2*(1+x)*(1+2x)
  2 + 10x + 16x^2 +  8x^3 = 2*(1+x)*(1+2x)^2
  2 + 14x + 36x^2 + 40x^3 + 16x^4 = 2*(1+x)*(1+2x)^3

Cf. A185045, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A185045 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208659 *)
(* Using the function RiordanSquare defined in A321620 we also have: *)
A208659 = RiordanSquare[(1 + x)/(1 - x), 16] // Flatten (* Gerry Martens, Oct 16 2022 *)

u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = u(n-1,x) + 2x*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
As triangle T(n,k) with 0 <= k <= n: T(n,k) = A029653(n,k)*2^k. - Philippe Deléham, Mar 04 2012
Sum_{k=0..n} T(n,k)*x^k = 2*(1+x)*(1+2x)^(n-2) for n > 1. - Philippe Deléham, Mar 05 2012

A321331 Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 14, 18, 4, 1, 30, 75, 40, 5, 1, 62, 270, 260, 75, 6, 1, 126, 903, 1400, 700, 126, 7, 1, 254, 2898, 6804, 5250, 1596, 196, 8, 1, 510, 9075, 31080, 34755, 15876, 3234, 288, 9, 1, 1022, 27990, 136420, 212625, 136962, 41160, 6000, 405, 10, 1, 2046, 85503, 583000, 1233650, 1076922, 447909, 95040, 10395, 550, 11

Offset: 0

Wolfdieter Lang, Dec 03 2018

This lower triangular matrix T is the inverse of the triangular matrix with elements Narumi[-1](n,m)/(m+1) = S1(n+1, m+1)/(n+1), with the Narumi triangle for parameter a = -1, and S1 = A048994 (Stirling1), i.e., Sum_{k=m..n} T(n, k) * S1(k+1, m+1)/(k+1) = delta_{n,m} (Kronecker symbol).
This triangle arises from the inverse of the rational Sheffer matrix Narumi[-1] = (log(1+x)/x, log(1+x) (such special Sheffer matrices (g(x), x*g(x)) define elements of the Narumi subgroup). The inverse matrix is (Narumi[-1])^(-1) = ((exp(x) - 1)/x, exp(x) - 1).
In order to have an integer matrix one takes T(n, k) := (n+1)*(Narumi[-1])^(-1)(n, k) = (k+1)*S2(n+1, k+1). The connection to S2 = A048993 results from the general relation between each Narumi-type matrix N = (g(x), x*g(x)) and its associated Sheffer matrix J = (1, x*g(x)) (this is of the Jabotinsky-type), i.e., N(n, m) = (m+1)*J(n+1, m+1)/(n+1), or with the row polynomials Npol(n, x) = (1/(n+1))*(d/dx)Jpol(n+1, x).
The signed triangle (-1)^(n-k)*A028421(n, k) (with upper diagonals filled with zeros) gives the integer matrix Nscaled with elements (n+1)*Narumi[-1](n,k). This inverse of Nscaled has the rational elements (Narumi[-1])^(-1)(k, m)/(m+1) = (1/(k+1))*S2(k+1, m+1).
The a- and z- sequence for the Sheffer matrix (Narumi[-1])^(-1) (see A006232 for a link on these sequences) have e.g.f.s Ea(x) = x/log(1 + x) and Ez(x) = 1/log(1 + x) - 1/x, hence a(n) = A006232(n)/A006233(n) and z(n) = A006232(n+1)/A075178(n), for n >= 0. This leads to the recurrence for T(n, k) given in the formula section.
The Boas-Buck-type column recurrence (see the link, also for references) uses the sequence with o.g.f. GBB(y) = exp(y)/(exp(y) - 1) - 1/y, with BB(n) = (-1)^(n+1)*A060054(n+1 ) / A227830(n+1), for n >= 1. For the recurrence see the formula section.
The Meixner-type identity (see the Meixner link) for the row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k, derived from the one for the Narumi[-1]^(-1) row polynomials is Sum_{k=1..n} (-1)^{k+1}*(1/k)*(d/dx)^k R(n, x)/(n+1) = R(n-1, x), for n >= 1, and R(0, x) = 1. Here d/dx is a differentiation operator.
The Roman-type recurrence for the row polynomials (see the reference, Corollary 3.7.2. p. 50) becomes, with the z-sequence from above: R(n, x) = ((n+1)/n)*{(x + 1/2)*1 + (x - z(1))*d/dx - Sum_{k=2..n-1} (1/k!)*z(k)*(d/dx)^k}*R(n-1, x), for n >= 1, and R(0, x) = 1.
The triangle is the exponential Riordan square (cf. A321620) of exp(x)-1 with an additional main diagonal of zeros. - Peter Luschny, Jan 03 2019

			The triangle T(n, k) begins:
  n\k  0    1     2      3       4       5      6     7     8   9 10 ...
  ----------------------------------------------------------------------
  0:   1
  1:   1    2
  2:   1    6     3
  3:   1   14    18      4
  4:   1   30    75     40       5
  5:   1   62   270    260      75       6
  6:   1  126   903   1400     700     126      7
  7:   1  254  2898   6804    5250    1596    196     8
  8:   1  510  9075  31080   34755   15876   3234   288     9
  9:   1 1022 27990 136420  212625  136962  41160  6000   405  10
  10:  1 2046 85503 583000 1233650 1076922 447909 95040 10395 550 11
  ...
Recurrence (from Stirling2): T(4, 2) = 3*(T(3, 2) + T(3, 1)/2) = 3*(18 + 14/2) = 75.
Recurrence (from a- and z-sequence): T(4, 0) = 5*((1/2)*T(3, 0) - (1/12)*T(3, 1) + (1/12)*T(3, 2) - (19/120)*T(3, 3)) = 5*(1/2 - 14/12 + 18/12 - 4*19/120) = 1; T(4,2) = (5/2)*(1*1*T(3, 1) + 2*(1/2)*T(3, 2) + 3*(-1/6)* T(3, 3)) = (5/2)*(14 + 18 - 2) = 75.
Recurrence for column k=2 (Boas-Buck-type): T(4, 2) = (5!*3/2)*((1/3!)*(1/12)*T(2, 2) + (1/4!)*(1/2)*T(3, 2)) = (5!*3/2)*((1/72)*3 + (1/48)*18) = 75.
Meixner identity for the row polynomials, for n = 3: {d/dx  - (1/2)*(d/dx)^2 + (1/3)*(d/dx)^3)}*R(3, x)/4) = ((14 - 36/2 + 24/3) + (36 - 24/2)*x + 12*x^2)/4 = (1 + 6*x + 3*x^2) = R(2, x).
Roman type recurrence for row polynomials: R(n, 3) = (3/2)*{(x + 1/12)*(1 + 6*x + 3*x^2) + (x - (-1/2))*(6 + 6*x) - (1/2!)*(1/12)*6} = 1 + 14*x + 18*x^2 + 4*x^3.

Steven Roman, The umbral calculus, Academic Press, 1984.

Wolfdieter Lang, On the Generating Functions of the Boas-Buck Sequences for the Inverse of Riordan and Sheffer Matrices
J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.

Cf. A006232, A006233, A060054, A028421, A048993, A048994, A075178, A227830.

Flat(List([0..10],n->List([0..n],k->(k+1)*Stirling2(n+1,k+1)))); # Muniru A Asiru, Dec 03 2018

T:=(n,k)->(k+1)*Stirling2(n+1,k+1): seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Dec 03 2018

T[n_, k_] := (k+1) * StirlingS2[n+1, k+1];  Table[T[n, k], {n,0,10}, {k, 0, n}] //Flatten (* Amiram Eldar, Dec 03 2018 *)

T(n, k) = (k+1)*stirling(n+1, k+1, 2) \\ Thomas Scheuerle, Nov 10 2023

# uses[riordan_square from A321620]
riordan_square(exp(x) - 1, 10, True) # Peter Luschny, Jan 03 2019

T(n, k) = (k+1)*A048993(n+1, k+1), with A048993 = Stirling2, for n >= k >= 0, and 0 otherwise.
T(n, k) = (n+1)*(Narumi[a=-1])^(-1)(n, k), with the Narumi[a=-1] matrix with entries (-1)^(n-k)*A028421(n, k)/(n+1).
E.g.f. for column k sequence: E(k, x) = (x*d/dx + 1)*EN(k, x), where EN(k, x) = (exp(x) - 1)^(k+1)/(x*k!) is the e.g.f. for the (Narumi[a=-1])^(-1) columns. Hence E(k, x) = exp(x)*(exp(x) - 1)*(k+1)/k!, for k >= 0.
E.g.f. for (ordinary) row polynomials R(n, x): Epol(z, x) = exp(z)*exp(x*(exp(z) - 1))*(1 + x*(exp(z) - 1)).
Recurrence (from Stirling2): T(n, k) = 0 for n < k; T(n, 0) = (k + 1)*T(n-1, k), for n <= 1, T(0, 0) = 1; T(n, k) = (k+1)*(T(n-1, k) + T(n-1, k-1)/k), for n >= 1, k >= 1.
Recurrence (from a- and z-sequence, see above): a = {1, 1/2, -1/6, 1/4, -19/30, 9/4, ...}, z = {1/2, -1/12, 1/12, -19/120, 9/20, -863/504, ...}.
T(n, k) = 0, for n < k; T(n, 0) = (n+1)*Sum_{j=0..n-1} z(j)*T(n-1, j), for n >= 1, with T(0, 0) = 1; T(n, k) = ((n+1)/k)*Sum_{j=0..n-m} binomial(k-1+j, j)*a(j)*T(n-1, k-1+j).
Recurrence for column k, from the Boas-Buck-type sequence BB(n) = (-1)^(n+1)*A060054(n+1)/A227830(n+1), for n >= 0; BB = {1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, ...}: T(n, k) = 0, for  n < k; T(n, n) = n+1, for n >= 0; T(n, k) = ((n+1)!*(k+1)/(n-k))*Sum_{j=k..n-1} (1/(j+1)!)*BB(n-(j+1))*T(j, k), for n >= 0 and k = 0, 1, ..., n-1.
T(n, k) = Stirling2(n+2, k+1) - Stirling2(n+1, k). - Peter Luschny, May 26 2020

A321966 Triangle read by rows, coefficients of a family of orthogonal polynomials, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 6, 27, 12, 1, 24, 168, 123, 22, 1, 120, 1200, 1275, 365, 35, 1, 720, 9720, 13950, 5655, 855, 51, 1, 5040, 88200, 163170, 87465, 18480, 1722, 70, 1, 40320, 887040, 2046240, 1387680, 383145, 49476, 3122, 92, 1

Offset: 0

Peter Luschny, Dec 20 2018

The polynomials represent a family of orthogonal polynomials which obey a recurrence of the form p(n, x) = (x + alpha(n))*p(n-1, x) - beta(n)*p(n-2, x) + gamma(n)*p(n-3, x). For the details see the Maple program.
We conjecture that the polynomials have only negative and simple real roots.
From Giuliano Cabrele, Sep 09 2021: (Start)
Let He(n,x) define the probabilist's version of Hermite polynomials.
Then the terms of the triangle appear to be the connection coefficients in
x^n*He(n,x) = Sum_{k=0..n} T(n,k)*He(2k,x).
These are generated by the explicit formula
T(n,m) = 2^(n-m)*Sum_{j=0..floor(n/2)} C(n,2*j)*C(2*n-2*j,2*m)*Gamma(1/2 + n - m - j)/Gamma(1/2 - j).
A formal proof that they correspond to the original definition is needed. (End)

			p(0,x) = 1;
p(1,x) = x + 1;
p(2,x) = x^2 + 5*x + 2;
p(3,x) = x^3 + 12*x^2 + 27*x + 6;
p(4,x) = x^4 + 22*x^3 + 123*x^2 + 168*x + 24;
p(5,x) = x^5 + 35*x^4 + 365*x^3 + 1275*x^2 + 1200*x + 120;
p(6,x) = x^6 + 51*x^5 + 855*x^4 + 5655*x^3 + 13950*x^2 + 9720*x + 720;

Peter Luschny, Plot of the polynomials
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 21.

p(n, 1) = A321965(n); p(n, 0) = n! = A000142(n).

Cf. A321620.

P := proc(n) option remember; local a, b, c;
a := n -> 3*n-2; b := n -> (n-1)*(3*n-4); c := n -> (n-2)^2*(n-1);
if n = 0 then return 1 fi;
if n = 1 then return x + 1 fi;
if n = 2 then return x^2 + 5*x + 2 fi;
expand((x+a(n))*P(n-1) - b(n)*P(n-2) + c(n)*P(n-3)) end:
seq(print(P(n)), n=0..6); # Computes the polynomials.

a[n_] := 3n-2; b[n_] := (n-1)(3n-4); c[n_] := (n-2)^2 (n-1);
P[n_] := P[n] = Switch[n, 0, 1, 1, x+1, 2, x^2 + 5x + 2, _, Expand[(x+a[n]) P[n-1] - b[n] P[n-2] + c[n] P[n-3]]];
Table[CoefficientList[P[n], x], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 01 2019, from Maple *)

# uses[RiordanSquare from A321620]
R = RiordanSquare((1 - 2*x)^(-1/2), 9, True).inverse()
for n in (0..8): print([(-1)^(n-k)*c for (k, c) in enumerate(R.row(n)[:n+1])])

Let R be the inverse of the Riordan square [see A321620] of (1 - 2*x)^(-1/2) then T(n, k) = (-1)^(n-k)*R(n, k).

A187889 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2+x^3)/(1-x-x^2-x^3)).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 7, 19, 18, 7, 1, 13, 43, 54, 32, 9, 1, 24, 94, 147, 117, 50, 11, 1, 44, 200, 375, 375, 216, 72, 13, 1, 81, 418, 913, 1100, 799, 359, 98, 15, 1, 149, 861, 2147, 3027, 2657, 1507, 554, 128, 17, 1, 274, 1753, 4914, 7937, 8174, 5610, 2603, 809, 162, 19, 1

Offset: 0

Emanuele Munarini, Mar 15 2011

			Triangle begins:
1
1,1
2,3,1
4,8,5,1
7,19,18,7,1
13,43,54,32,9,1
24,94,147,117,50,11,1
44,200,375,375,216,72,13,1
81,418,913,1100,799,359,98,15,1

Cf. A104580, A321620.

(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1/(1 - x - x^2- x^3), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(sum(binomial(i+k,k)*trinomial(i+k,n-k-i),i,0,n-k),n,0,8,k,0,n);

a(n,k) = Sum_{i=0..n-k} binomial(i+k,k)*trinomial(i+k,n-k-i), where trinomial(n,k) are the trinomial coefficients (A027907).

Recurrence: a(n+3,k+1) = a(n+2,k+1) + a(n+2,k) + a(n+1,k+1) + a(n+1,k) + a(n,k+1) + a(n,k)

A321623 The Riordan square of the large Schröder numbers, triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 2, 2, 6, 10, 4, 22, 46, 32, 8, 90, 214, 196, 88, 16, 394, 1018, 1104, 672, 224, 32, 1806, 4946, 6020, 4448, 2048, 544, 64, 8558, 24470, 32400, 27432, 15584, 5792, 1280, 128, 41586, 122926, 173572, 162680, 107408, 49824, 15552, 2944, 256

Offset: 0

Peter Luschny, Nov 22 2018

Triangle, read by rows,given by [2,1,2,1,2,1,2,1,...]DELTA[2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 05 2020

			[0][     1]
[1][     2,      2]
[2][     6,     10,      4]
[3][    22,     46,     32,      8]
[4][    90,    214,    196,     88,     16]
[5][   394,   1018,   1104,    672,    224,     32]
[6][  1806,   4946,   6020,   4448,   2048,    544,     64]
[7][  8558,  24470,  32400,  27432,  15584,   5792,   1280,   128]
[8][ 41586, 122926, 173572, 162680, 107408,  49824,  15552,  2944,  256]
[9][206098, 625522, 929248, 942592, 697408, 379840, 149248, 40192, 6656, 512]

T(n, 0) = A006318 (large Schröder), A321574 (row sums), A000007 (alternating row sums).

Cf. A321620, A133367.

# The function RiordanSquare is defined in A321620.
LargeSchröder := x -> (1 - x - sqrt(1 - 6*x + x^2))/(2*x);
RiordanSquare(LargeSchröder(x), 10);

(* The function RiordanSquare is defined in A321620. *)
LargeSchröder[x_] := (1 - x - Sqrt[1 - 6*x + x^2])/(2*x);
RiordanSquare[LargeSchröder[x], 10] (* Jean-François Alcover, Jun 15 2019, from Maple *)

# uses[riordan_square from A321620]
riordan_square((1 - x - sqrt(1 - 6*x + x^2))/(2*x), 10)

T(n, k) = 2^k*A133367(n,k). - Philippe Deléham, Feb 05 2020

cp's OEIS Frontend

A116392 Riordan array (1/sqrt(1-2*x-3*x^2), 1/sqrt(1-2*x-3*x^2) -1).

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A132372 T(n, k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k. Triangle T(n,k), read by rows.

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A109956 Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.

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A154380 The Riordan square of the Bell numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

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A172094 The Riordan square of the little Schröder numbers A001003.

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A208659 Triangle of coefficients of polynomials v(n,x) jointly generated with A185045; see the Formula section.

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A321331 Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).

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A116392 Riordan array (1/sqrt(1-2x-3x^2), 1/sqrt(1-2x-3x^2) -1).