A256893 -id:A256893 - cp's OEIS frontend

A122832 Exponential Riordan array (e^(x(1+x)),x).

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 25, 28, 18, 4, 1, 81, 125, 70, 30, 5, 1, 331, 486, 375, 140, 45, 6, 1, 1303, 2317, 1701, 875, 245, 63, 7, 1, 5937, 10424, 9268, 4536, 1750, 392, 84, 8, 1, 26785, 53433, 46908, 27804, 10206, 3150, 588, 108, 9, 1

Offset: 0

Paul Barry, Sep 12 2006

Row sums are A000898. Inverse is A122833. Product of A007318 and A067147.

			Triangle begins:
   1;
   1,   1;
   3,   2,  1;
   7,   9,  3,  1;
  25,  28, 18,  4, 1;
  81, 125, 70, 30, 5, 1;
  ...
From _Peter Bala_, May 14 2012: (Start)
T(3,1) = 9. The 9 ways to select a subset of {1,2,3} of size 1 and arrange the remaining elements into a set of lists (denoted by square brackets) of length 1 or 2 are:
{1}[2,3], {1}[3,2], {1}[2][3],
{2}[1,3], {2}[3,1], {2}[1][3],
{3}[1,2], {3}[2,1], {3}[1][2]. (End)

Michel Marcus, Rows n=0..50 of triangle, flattened

A000898 (row sums), A047974 (column 0), A291632 (column 1), A122833 (inverse array).

Cf. A059344, A111062.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

T(n,k) = (n!/k!)*sum(i=0, n-k, binomial(i,n-k-i)/i!); \\ Michel Marcus, Aug 28 2017

Number triangle T(n,k) = (n!/k!)*Sum_{i = 0..n-k} C(i,n-k-i)/i!.
From Peter Bala, May 14 2012: (Start)
Array is exp(S + S^2) where S is A132440 the infinitesimal generator for Pascal's triangle.
T(n,k) = binomial(n,k)*A047974(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then arrange the remaining n-k elements into a set of lists of length 1 or 2. (End)
From Peter Bala, Oct 24 2023: (Start)
n-th row polynomial: R(n,x) = exp(D + D^2) (x^n) = exp(D^2) (1 + x)^n, where D denotes the derivative operator d/dx. Cf. A111062.
The sequence of polynomials defined by R(n,x-1) = exp(D^2) (x^n) begins [1, 1, 2 + x^2, 6*x + x^3, 12 + 12*x^2 + x^4, ...] and is related to the Hermite polynomials. See A059344. (End)

More terms from Michel Marcus, Aug 28 2017

A147746 Riordan array (1, x(1-2x)/(1-3x+x^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 13, 14, 9, 4, 1, 0, 34, 40, 28, 14, 5, 1, 0, 89, 114, 87, 48, 20, 6, 1, 0, 233, 323, 267, 161, 75, 27, 7, 1, 0, 610, 910, 809, 528, 270, 110, 35, 8, 1

Offset: 0

Paul Barry, Nov 11 2008

Triangle [0,1,1,1,0,0,0,....] DELTA [1,0,0,0,...] with Deléham DELTA as in A084938.
Note that 1/(1-x/(1-x/(1-x))) = (1-2x)/(1-3x+x^2). Row sums are A124302.
A147746*A007318 = A147747.

			Triangle begins
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   5,   5,   3,   1;
  0,  13,  14,   9,   4,   1;
  0,  34,  40,  28,  14,   5,   1;
  0,  89, 114,  87,  48,  20,   6,   1;
  ...

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, # (1-2#)/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Sum_{k=0..n} T(n,k)*2^k = A147748(n). - Philippe Deléham, Oct 30 2011
Sum_{k=0..n} T(n,k)*(-1)^(n-k) = A215936(n). - Philippe Deléham, Aug 30 2012
G.f.: (1 - 3*x + x^2)/(1 - 3*x + x^2 - x*y + 2*x^2*y). - R. J. Mathar, Aug 11 2015

A154602 Exponential Riordan array [exp(sinh(x)exp(x)), sinh(x)exp(x)].

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 49, 104, 70, 16, 1, 257, 641, 550, 190, 25, 1, 1539, 4380, 4531, 2080, 425, 36, 1, 10299, 32803, 39515, 22491, 6265, 833, 49, 1, 75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1, 609441, 2337505, 3575820, 2792476, 1192086, 281190, 36204, 2460, 81, 1

Offset: 0

Paul Barry, Jan 12 2009

Triangle T(n,k), read by rows, given by [1,2,1,4,1,6,1,8,1,10,1,12,1,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 02 2009

			Triangle begins
     1;
     1,    1;
     3,    4,    1;
    11,   19,    9,    1;
    49,  104,   70,   16,   1;
   257,  641,  550,  190,  25,  1;
  1539, 4380, 4531, 2080, 425, 36, 1;
Production matrix of this array is
  1,  1,
  2,  3,  1,
  0,  4,  5,  1,
  0,  0,  6,  7,  1,
  0,  0,  0,  8,  9,  1,
  0,  0,  0,  0, 10, 11,  1
with generating function exp(t*x)*(1+t)*(1+2*x).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Columns k=0..3 give A000007, A383203, A383204, A383205.
Cf. A004211 (first column), A256893.
Sums include: A000007 (alternating sign row), A055882 (row sums).

A154602:= func< n,k | (&+[2^(n-j)*Binomial(j,k)*StirlingSecond(n,j): j in [k..n]]) >;
[A154602(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 19 2024

A154602 := (n, k) -> add(2^(n-j) * binomial(j, k) * Stirling2(n, j), j = k..n): for n from 0 to 6 do seq(A154602(n, k), k = 0..n) od; # Peter Luschny, Dec 13 2022

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[Sinh[#] Exp[#]]&, Sinh[#] Exp[#]&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

def A154602(n,k): return sum(2^(n-j)*binomial(j,k)* stirling_number2(n,j) for j in range(k,n+1))
flatten([[A154602(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 19 2024

T(n, 0) = A004211(n).
Sum_{k=0..n} T(n, k) = A055882(n) (row sums).
From Peter Bala, Jun 15 2009: (Start)
T(n,k) = Sum_{i = k..n} 2^(n-i)*binomial(i,k)*Stirling2(n,i).
E.g.f.: exp((t+1)/2*(exp(2*x)-1)) = 1 + (1+t)*x + (3+4*t+t^2)*x^2/2! + ....
Row generating polynomials R_n(x):
R_n(x) = 2^n*Bell(n,(x+1)/2), where Bell(n,x) = Sum_{k = 0..n} Stirling2(n, k)*x^k denotes the n-th Bell polynomial.
Recursion:
R(n+1,x) = (x+1)*(R_n(x) + 2*d/dx(R_n(x))).
(End)
Recurrence: T(n,k) = 2*(k+1)*T(n-1,k+1) + (2*k+1)*T(n-1,k) + T(n-1,k-1). - Emanuele Munarini, Apr 14 2020
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n). - G. C. Greubel, Sep 19 2024
E.g.f. of column k (with leading zeros): f(x)^k * exp(f(x)) / k! with f(x) = (exp(2*x) - 1)/2. - Seiichi Manyama, Apr 19 2025

A106509 Riordan array ((1+x)/(1+x+x^2), x/(1+x)), read by rows.

Original entry on oeis.org

1, 0, 1, -1, -1, 1, 1, 0, -2, 1, 0, 1, 2, -3, 1, -1, -1, -1, 5, -4, 1, 1, 0, 0, -6, 9, -5, 1, 0, 1, 0, 6, -15, 14, -6, 1, -1, -1, 1, -6, 21, -29, 20, -7, 1, 1, 0, -2, 7, -27, 50, -49, 27, -8, 1, 0, 1, 2, -9, 34, -77, 99, -76, 35, -9, 1, -1, -1, -1, 11, -43, 111, -176, 175, -111, 44, -10, 1

Offset: 0

Paul Barry, May 04 2005

Row sums are A106510.
Diagonal sums are A106511.
Inverse of A072405 (when this starts 1, 0, 1, ...).

			Triangle begins:
   1;
   0,  1;
  -1, -1,  1;
   1,  0, -2,  1;
   0,  1,  2, -3,  1;
  -1, -1, -1,  5, -4,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Cf. A072405, A106510, A106511.

T:= func< n,k | (&+[ (-1)^j*Binomial(2*n-k-j, j): j in [0..n-k]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 + #)/(1 + # + #^2)&, #/(1 + #)&, 12] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

def T(n,k): return sum( (-1)^j*binomial(2*n-k-j, j) for j in (0..n-k))
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021

T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j).
T(n,k) = T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) - 2*T(n-2,k) + T(n-3,k-1) - T(n-3,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = 0, T(2,1) = T(2,0) = -1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014
Sum_{k=0..n} T(n,k) = A106510(n). - G. C. Greubel, Apr 28 2021

A112519 Riordan array (1, xc(x)c(-x*c(x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 4, 0, 1, 0, 12, 2, 6, 0, 1, 0, 14, 28, 3, 8, 0, 1, 0, 100, 32, 48, 4, 10, 0, 1, 0, 180, 249, 54, 72, 5, 12, 0, 1, 0, 990, 440, 455, 80, 100, 6, 14, 0, 1, 0, 2310, 2552, 792, 726, 110, 132, 7, 16, 0, 1, 0, 10920, 5876, 4836, 1248, 1070, 144, 168, 8, 18, 0, 1

Offset: 0

Paul Barry, Sep 09 2005

Row sums are A112520. Second column is essentially A055392. Inverse is A112517. Riordan array product (1, x*c(x))*(1, x*c(-x)).

			Triangle begins
  1;
  0,   1;
  0,   0,   1;
  0,   2,   0,   1;
  0,   1,   4,   0,  1;
  0,  12,   2,   6,  0,   1;
  0,  14,  28,   3,  8,   0,  1;
  0, 100,  32,  48,  4,  10,  0,  1;
  0, 180, 249,  54, 72,   5, 12,  0, 1;
  0, 990, 440, 455, 80, 100,  6, 14, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000108, A055392, A112517, A112520.

A112519:= func< n,k | n eq 0 and k eq 0 select 1 else (k/n)*(&+[(-1)^j*Binomial(2*n-k-j-1, n-k-j)*Binomial(2*j+k-1, j): j in [0..n-k]]) >;
[A112519(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 12 2022

(* First program *)
c[x_]:= (1 - Sqrt[1-4x])/(2x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, # c[#] c[-# c[#]]&, 12] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
(* Second program *)
T[n_, k_]:= If[k==n, 1, (k/n)*Binomial[2*n-k-1, n-1]*HypergeometricPFQ[{k-n, k/2, (1+k)/2}, {k-2*n+1, k}, -4]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 12 2022 *)

@CachedFunction
def A112519(n,k):
    if (k==n): return 1
    else: return (k/n)*sum( (-1)^j*binomial(2*n-k-j-1, n-k-j)*binomial(2*j+k-1, j) for j in (0..n-k) )
flatten([[A112519(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 12 2022

Riordan array (1, (sqrt(3-2*sqrt(1-4*x)) - 1)/2).
T(n, k) = (k/n)*Sum_{j=0..n} (-1)^(j-k)*C(2*n-j-1, n-j)*C(2*j-k-1, j-k), with T(0, 0) = 1.
T(n, k) = (k/n)*binomial(2*n-k-1, n-1)*Hypergoemetric3F2([k-n, k/2, (1+k)/2], [k-2*n+1, k], -4), with T(0, 0) = 1. - G. C. Greubel, Jan 12 2022

A113278 Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 3, -3, 3, 1, -15, 12, -6, 4, 1, 105, -75, 30, -10, 5, 1, -945, 630, -225, 60, -15, 6, 1, 10395, -6615, 2205, -525, 105, -21, 7, 1, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1, 2027025, -1216215, 374220, -79380, 13230, -1890, 252, -36, 9, 1

Offset: 0

Paul D. Hanna, Oct 22 2005

			Triangle begins:
  1;
  1,1;
  -1,2,1;
  3,-3,3,1;
  -15,12,-6,4,1;
  105,-75,30,-10,5,1;
  -945,630,-225,60,-15,6,1;
  10395,-6615,2205,-525,105,-21,7,1;
  ...
where T(n,k) = (-1)^(n-1-k)*A001147(n-1)*C(n,k).
The matrix square equals:
  1;
  2,1;
  0,4,1;
  0,0,6,1;
  0,0,0,8,1;
  0,0,0,0,10,1;
  0,0,0,0,0,12,1;
  ...
The matrix log, L, begins:
  0;
  1,0;
  -2,2,0;
  8,-6,3,0;
  -48,32,-12,4,0;
  384,-240,80,-20,5,0;
  -3840,2304,-720,160,-30,6,0;
  ...
where L(n,k) = (-1)^(n-1-k)*A000165(n-1)*C(n,k).

Peter Bala, Generalized Dobinski formulas

Cf. A039683, A122848.

Cf. A001147 (odd double factorials), A000165 (even double factorials).

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Sqrt[1 + 2 #]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r==c,1,if(r==c+1,2*c)))); (sum(i=0,n+1,(sum(j=1,n+1,-(M^0-M)^j/j)/2)^i/i!))[n+1,k+1]}

Exponential Riordan array [sqrt(1 + 2*x),x] with e.g.f. sqrt(1+2*x)*exp(t*x) = 1 + (1+t)*x + (-1+2*t+t^2)*x^2/2! + ... . The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*sum {k = 0..inf} (2*k+1)*(2*k-1)*...*(2*k+1-2*(n-1))*(x/2)^k/k!. Cf. A122848. - Peter Bala, Jun 23 2014

A114189 Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -13, 19, -7, 1, 67, -102, 44, -10, 1, -381, 593, -278, 78, -13, 1, 2307, -3640, 1795, -568, 121, -16, 1, -14589, 23231, -11849, 4051, -999, 173, -19, 1, 95235, -152650, 79750, -28770, 7820, -1598, 234, -22, 1, -636925, 1025965, -545680, 204760, -59650, 13642, -2392, 304, -25, 1

Offset: 0

Paul Barry, Nov 16 2005

Inverse of A114188. Factors as (1,xc(-2x))*(1/(1+x), x/(1+x)). Row sums are 0^n. Diagonal sums are A114190. First column is A114191. A signed version of A110506.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -1,-2,-2,-2,-2,-2,-2,...] DELTA [1,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 01 2007

			Triangle begins
     1;
    -1,    1;
     3,   -4,    1;
   -13,   19,   -7,   1;
    67, -102,   44, -10,   1;
  -381,  593, -278,  78, -13, 1;

c[x_] := (1 - Sqrt[1 - 4x])/(2x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 + # c[-2#])&, # c[-2#]/(1 + # c[-2#])&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Riordan array ((3-sqrt(1+8x))/(2(1-x)), (sqrt(1+8x)-2x-1)/(2(1-x))).

T(n,k) = (-1)^(n-k)*A110506(n,k). - Philippe Deléham, Mar 24 2007

A114193 Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -2, 1, 8, -6, 1, -40, 36, -10, 1, 224, -224, 80, -14, 1, -1344, 1440, -600, 140, -18, 1, 8448, -9504, 4400, -1232, 216, -22, 1, -54912, 64064, -32032, 10192, -2184, 308, -26, 1, 366080, -439296, 232960, -81536, 20160, -3520, 416, -30, 1, -2489344, 3055104, -1697280, 639744, -176256, 35904, -5304, 540, -34, 1

Offset: 0

Paul Barry, Nov 16 2005

Row sums are A114191. Diagonal sums are A114194. Inverse of A114192.

Triangle T(n,k), read by rows, given by (-2, -2, -2, -2, -2, -2, -2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 26 2014

			Triangle begins
      1;
     -2,    1;
      8,   -6,    1;
    -40,   36,  -10,   1;
    224, -224,   80, -14,   1;
  -1344, 1440, -600, 140, -18, 1;

Cf. A039599, A084938.

c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 + 2 # c[-2 #])&, # c[-2 #]/(1 + 2 # c[-2 #])&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Riordan array ((sqrt(1+8x)-1)/(4x), (sqrt(1+8x)-1)^2/(16x)).

T(n, k) = (-2)^(n-k)*A039599(n, k) = (-2)^(n-k)*C(2*n, n-k)*(2*k+1)/(n+k+1). - Philippe Deléham, Nov 17 2005

A116071 Triangle T, read by rows, equal to Pascal's triangle to the matrix power of Pascal's triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 41, 40, 18, 4, 1, 196, 205, 100, 30, 5, 1, 1057, 1176, 615, 200, 45, 6, 1, 6322, 7399, 4116, 1435, 350, 63, 7, 1, 41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1, 293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1

Offset: 0

Paul D. Hanna, Feb 03 2006

Column 0 = A000248 (Number of forests with n nodes and height at most 1).
Column 1 = A052512 (Number of labeled trees of height 2).
Row sums = A080108 (Sum_{k=1..n} k^(n-k) * C(n-1,k-1)).
Central terms = A116072(n) = (n+1) * A000108(n) * A000248(n).
From Peter Bala, Sep 13 2012: (Start)
For commuting lower unitriangular matrix A and lower triangular matrix B we define A raised to the matrix power B, denoted by A^B, to be the lower unitriangular matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential defined by the power series
  Exp(A) = 1 + A + A^2/2! + A^3/3! + ...
and the matrix logarithm Log(A) is defined by the series
  Log(A) = (A-1) - 1/2*(A-1)^2/2 + 1/3*(A-1)^3 - ....
Let A = [f(x),x] and B = [g(x),x] be exponential Riordan arrays in the Appell subgroup and suppose f(0) = 1. Then A and B commute and A^B is the exponential Riordan array [exp(g(x)*log(f(x))),x], also belonging to the Appell group. In the present case we are taking A = B = [exp(x),x], equal to the Pascal triangle A007318.
For any lower unitriangular matrix A (with, say, rational entries) the infinite tower of powers A^(A^(A^(...))) is well-defined (and also has rational entries). An example is given in the Formula section. (End)

			E.g.f.: E(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2/2!
  + (10 + 9*y + 3*y^2 + y^3)*x^3/3!
  + (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4/4!
  + (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5/5! +...
where E(x,y) = exp(x*y) * exp(x*exp(x)).
O.g.f.: A(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2
  + (10 + 9*y + 3*y^2 + y^3)*x^3
  + (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4
  + (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5 +...
where
A(x,y) = 1/(1-x*y) + x/(1-x*(y+1))^2 + x^2/(1-x*(y+2))^3 + x^3/(1-x*(y+3))^4 + x^4/(1-x*(y+4))^5 + x^5/(1-x*(y+5))^6 + x^6/(1-x*(y+6))^7 + x^7/(1-x*(y+7))^8 +...
Triangle begins:
  1;
  1, 1;
  3, 2, 1;
  10, 9, 3, 1;
  41, 40, 18, 4, 1;
  196, 205, 100, 30, 5, 1;
  1057, 1176, 615, 200, 45, 6, 1;
  6322, 7399, 4116, 1435, 350, 63, 7, 1;
  41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1;
  293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080 (rows 0..45 of flattened triangle).

Cf. A000248, A052512, A080108, A116072, A215652.

Cf. A080108, A216689, A240165, A245834, A245835, A216973.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[# Exp[#]]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

/* By definition C^C: */
{T(n,k)=local(A, C=matrix(n+1,n+1,r,c,binomial(r-1,c-1)), L=matrix(n+1,n+1,r,c,if(r==c+1,c))); A=sum(m=0,n,L^m*C^m/m!); A[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From e.g.f.: */
{T(n,k)=local(A=1);A=exp( x*y + x*exp(x +x*O(x^n)) );n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From o.g.f. (Paul D. Hanna, Aug 03 2014): */
{T(n,k)=local(A=1);A=sum(k=0, n, x^k/(1 - x*(k+y) +x*O(x^n))^(k+1));polcoeff(polcoeff(A, n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From row polynomials (Paul D. Hanna, Aug 03 2014): */
{T(n,k)=local(R);R=sum(k=0,n,(k+y)^(n-k)*binomial(n,k));polcoeff(R,k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From formula for T(n,k) (Paul D. Hanna, Aug 03 2014): */
{T(n,k) = sum(j=0,n-k, binomial(n,j) * binomial(n-j,k) * j^(n-k-j))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

E.g.f.: exp( x*exp(x) + x*y ).
From Peter Bala, Sep 13 2012: (Start)
Exponential Riordan array [exp(x*exp(x)),x] belonging to the Appell group. Thus the e.g.f. for the k-th column of the triangle is x^k/k!*exp(x*exp(x)).
The inverse array, denote it by X, is a signed version of A215652. The infinite tower of matrix powers X^(X^(X^(...))) equals the inverse of Pascal's triangle. (End)
O.g.f.: Sum_{n>=0} x^n / (1 - x*(n+y))^(n+1). - Paul D. Hanna, Aug 03 2014
G.f. for row n: Sum_{k=0..n} binomial(n,k) * (k + y)^(n-k) for n>=0. - Paul D. Hanna, Aug 03 2014
T(n,k) = Sum_{j=0..n-k} C(n,j) * C(n-j,k) * j^(n-k-j) = A000248(n-k)*C(n,k). - Paul D. Hanna, Aug 03 2014
Infinitesimal generator is A216973. - Peter Bala, Feb 13 2017

A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 80, 30, 6, 1, 880, 320, 60, 8, 1, 12320, 4400, 800, 100, 10, 1, 209440, 73920, 13200, 1600, 150, 12, 1, 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1, 96342400, 33510400, 5864320, 689920, 61600, 4480, 280, 16, 1

Offset: 0

Paul D. Hanna, Feb 07 2008

This array is the particular case P(2,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown in the comments to A094587. - Peter Bala, Jul 10 2008

The row polynomials form an Appell sequence. - Tom Copeland, Dec 03 2013

			Triangle begins:
1;
2, 1;
10, 4, 1;
80, 30, 6, 1;
880, 320, 60, 8, 1;
12320, 4400, 800, 100, 10, 1;
209440, 73920, 13200, 1600, 150, 12, 1;
4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ...

Wikipedia, Appell sequence
Wikipedia, Sheffer sequence

Cf. A136215 (square-root), A112333, A008544, A136212, A136213.

Cf. A094587.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{T(n,k) = binomial(n,k)*if(n-k==0,1, prod(j=0,n-k-1,3*j+2))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333.
Equals the matrix square of triangle A136215.
T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - Peter Bala, Jul 10 2008
Using the formalism of A132382 modified for the triple rather than the double factorial (replace 2 by 3 in basic formulas), the e.g.f. for the row polynomials is exp(x*t)*(1-3x)^(-2/3). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 28 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(2/3), y].
The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (2*x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (80, 30, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 5)*(2*x + 8) = 80 + 20*(2*x) + 6*(2*x*(2*x + 3)) + (2*x)*(2*x + 3)*(2*x + 6). (End)

cp's OEIS Frontend

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A106509 Riordan array ((1+x)/(1+x+x^2), x/(1+x)), read by rows.

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A112519 Riordan array (1, x*c(x)*c(-x*c(x))), c(x) the g.f. of A000108.

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A113278 Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0.

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A114189 Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))), c(x) the g.f. of A000108.

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A154602 Exponential Riordan array [exp(sinh(x)exp(x)), sinh(x)exp(x)].

A112519 Riordan array (1, xc(x)c(-x*c(x))), c(x) the g.f. of A000108.