A008297 -id:A008297 - cp's OEIS frontend

A049410 A triangle of numbers related to triangle A049325.

1, 3, 1, 6, 9, 1, 6, 51, 18, 1, 0, 210, 195, 30, 1, 0, 630, 1575, 525, 45, 1, 0, 1260, 10080, 6825, 1155, 63, 1, 0, 1260, 51660, 71505, 21840, 2226, 84, 1, 0, 0, 207900, 623700, 333585, 57456, 3906, 108, 1, 0, 0, 623700, 4573800, 4293135, 1195425, 131670

Offset: 1

Wolfdieter Lang

a(n,1)= A008279(3,n-1). a(n,m)=: S1(-3; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m)= A008275 (signed Stirling first kind), S1(2; n,m)= A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A000369(n,m).
The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins:
  {1};
  {3,1};
  {6,9,1};
  {6,51,18,1};
  ...
E.g. row polynomial E(3,x)= 6*x+9*x^2+x^3.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Row sums give A049426.

rows = 10;
t = Table[Product[4k+3, {k, 0, n-1}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
M = Inverse[Array[T, {rows, rows}]] // Abs;
A049325 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

# uses[inverse_bell_transform from A265605]
# Adds a column 1,0,0,0,... at the left side of the triangle.
multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))
inverse_bell_matrix(multifact_4_3, 9) # Peter Luschny, Dec 31 2015

a(n, m) = n!*A049325(n, m)/(m!*4^(n-m)); a(n, m) = (4*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

Original entry on oeis.org

1, 2, 9, 58, 485, 4986, 60877, 861554, 13878153, 250854130, 5030058161, 110837000682, 2662669300909, 69270266115818, 1940260799150325, 58220372514830626, 1863293173842259217, 63356877145370671074

Offset: 0

Vladeta Jovovic, Jan 06 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			(1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for sequences related to Laguerre polynomials

Cf. A001861, A049020, A052897, A059099, A059110.

Row sums of A059114.

[Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

s := series((1-x)/(1-2*x)*exp(x/(1-x)), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2020 *)
Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

{a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* Michael Somos, Aug 03 2006 */

{a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* Michael Somos, Aug 03 2006 */

a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ Michel Marcus, Feb 23 2021

[factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).
Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - Michael Somos, Aug 03 2006
a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - G. C. Greubel, Feb 23 2021
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Feb 23 2021

Definition clarified by Harvey P. Dale, Jul 18 2020

A134146 Triangle of numbers obtained from the partition array A134145.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 24, 3, 1, 945, 150, 24, 3, 1, 10395, 1485, 177, 24, 3, 1, 135135, 14805, 1620, 177, 24, 3, 1, 2027025, 191520, 16425, 1701, 177, 24, 3, 1, 34459425, 2687580, 208125, 16830, 1701, 177, 24, 3, 1, 654729075, 44552025, 2880360, 212985

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(3)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [3,1]; [15,3,1]; [105,24,3,1]; [945,150,24,3,1];...

W. Lang, First 10 rows and more.

Cf. A134147 (row sums).
Cf. A134148 (allternating row sums).
Cf. A134134 (k=2 member of this triangle family).

a(n,m)=sum(product(S2(3;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(3;j,1)= A001147(j) = A035342(j,1) = (2*j-1)!!.

A134275 Triangle of numbers obtained from the partition array A134274.

Original entry on oeis.org

1, 5, 1, 45, 5, 1, 585, 70, 5, 1, 9945, 810, 70, 5, 1, 208845, 14895, 935, 70, 5, 1, 5221125, 284895, 16020, 935, 70, 5, 1, 151412625, 7055100, 309645, 16645, 935, 70, 5, 1, 4996616625, 192734100, 7526475, 315270, 16645, 935, 70, 5, 1, 184874815125

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(5)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			Triangle begins:
  [1];
  [5,1];
  [45,5,1];
  [585,70,5,1];
  [9945,810,70,5,1];
  ...

Wolfdieter Lang, First 10 rows and more.

Cf. A134276 (row sums). A134277 (alternating row sums).

Cf. A134151 (S2(4)').

a(n,m) = sum(product(S2(5;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(5;j,1)= A007696(j) = A049029(j,1) = (4*j-3)(!^4), (quadruple- or 4-factorials).

A176013 Triangle, read by rows, T(n, k) = (-1)^n * n!/(kk!) binomial(n-1, k-1) * binomial(n, k-1).

Original entry on oeis.org

-1, 2, 1, -6, -9, -1, 24, 72, 24, 1, -120, -600, -400, -50, -1, 720, 5400, 6000, 1500, 90, 1, -5040, -52920, -88200, -36750, -4410, -147, -1, 40320, 564480, 1317120, 823200, 164640, 10976, 224, 1, -362880, -6531840, -20321280, -17781120, -5334336, -592704, -24192, -324, -1

Offset: 1

Roger L. Bagula, Apr 06 2010

Row sums are: -1, 3, -16, 121, -1171, 13711, -187468, 2920961, -50948677, 981458011, ...

			Triangle begins as:
       -1;
        2,        1;
       -6,       -9,        -1;
       24,       72,        24,         1;
     -120,     -600,      -400,       -50,       -1;
      720,     5400,      6000,      1500,       90,       1;
    -5040,   -52920,    -88200,    -36750,    -4410,    -147,     -1;
    40320,   564480,   1317120,    823200,   164640,   10976,    224,    1;
  -362880, -6531840, -20321280, -17781120, -5334336, -592704, -24192, -324, -1;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A008297.

[(-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 08 2021

T[n_, k_] = (-1)^n*n!/(k*k!)*Binomial[n-1, k-1]*Binomial[n, k-1];
Table[T[n, k], {n,12}, {k,n}]//Flatten

flatten([[(-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 08 2021

T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1).

T(n, k) = binomial(n+1, k) * A008297(n, k)/(n+1). - G. C. Greubel, Feb 08 2021

Edited by G. C. Greubel, Feb 08 2021

A048786 Triangle of coefficients of certain exponential convolution polynomials.

Original entry on oeis.org

1, 8, 1, 96, 24, 1, 1536, 576, 48, 1, 30720, 15360, 1920, 80, 1, 737280, 460800, 76800, 4800, 120, 1, 20643840, 15482880, 3225600, 268800, 10080, 168, 1, 660602880, 578027520, 144506880, 15052800, 752640, 18816, 224, 1

Offset: 1

Wolfdieter Lang

i) p(n,x) := sum(a(n,m)*x^m,m=1..n), p(0,x) := 1, are monic polynomials satisfying p(n,x+y)= sum(binomial(n,k)*p(k,x)*p(n-k,y),k=0..n), (exponential convolution polynomials). ii) In the terminology of the umbral calculus (see reference) p(n,x) are called associated to f(t)= t/(1+4*t). iii) a(n,1)= A034177(n).
Also the Bell transform of A034177. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
Also the fourth power of the unsigned Lah triangular matrix A105278. - Shuhei Tsujie, May 18 2019
Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -3 <= d <= 4). - Shuhei Tsujie, May 18 2019

			Triangle begins:
      1;
      8,     1;
     96,    24,    1;
   1536,   576,   48,  1;
  30720, 15360, 1920, 80, 1;
  ...

S. Roman, The Umbral Calculus, Academic Press, New York, 1984

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A034177, A038231, A008297.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 4^n*(n+1)!, 9); # Peter Luschny, Jan 28 2016

rows = 8;
t = Table[4^n*(n+1)!, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = n!*4^(n-m)*binomial(n-1, m-1)/m!, n >= m >= 1; a(n, m) := 0, m>n; a(n, m) = (n!/m!)*A038231(n-1, m-1) = 4^(n-m)*A008297(n, m) (Lah-triangle).

T(8,4) corrected by Jean-François Alcover, Jun 22 2018

A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 13, 21, 18, 6, 73, 136, 156, 96, 24, 501, 1045, 1460, 1260, 600, 120, 4051, 9276, 15030, 16320, 11160, 4320, 720, 37633, 93289, 170142, 219450, 192360, 108360, 35280, 5040, 394353, 1047376, 2107448, 3116736, 3294480, 2405760, 1149120, 322560, 40320

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			Triangle begins as:
    1;
    1,    1;
    3,    4,    2;
   13,   21,   18,    6;
   73,  136,  156,   96,  24;
  501, 1045, 1460, 1260, 600, 120;
  ...;
E.g.f. for T(n, 2) = (x/(1-x))^2*e^(x/(x-1)) = x^2 + 3*x^3 + 13/2*x^4 + 73/6*x^5 + 167/8*x^6 + 4051/120*x^7 + ...

Cf. A000262, A052852, A052897, A059110.

Row sums give A059115. Alternating row sums give A288268.

[Factorial(n)*Evaluate(LaguerrePolynomial(n-k, k-1), -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Table[n!*LaguerreL[n-k, k-1, -1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

T(n, k) = n! * pollaguerre(n-k, k-1, -1); \\ Michel Marcus, Feb 23 2021

flatten([[factorial(n)*gen_laguerre(n-k, k-1, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for T(n, k) = (x/(1-x))^k * exp(x/(x-1)).
T(n, k)= Sum_{i=0..n} L'(n,i) * ( Product_{j=1..k} (i-j+1) ).
T(n, 0) = A000262(n).
T(n, 1) = A052852(n).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = n! * k! * Sum_{j=0..n} binomial(j, k)*binomial(n-1, j-1)/j!.
T(n, k) = n! * Laguerre(n-k, k-1, -1).
T(n, k) = n!*binomial(n-1, k-1)*Hypergeometric1F1([k-n], [k], -1) with T(n, 0) = Hypergeometric2F0([1-n, -n], [], 1). (End)

A111597 Lah numbers: a(n) = n!*binomial(n-1,6)/7!.

Original entry on oeis.org

1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480, 779155977600, 20777492736000, 565147802419200, 15721384321843200, 448059453172531200, 13097122477350912000, 392913674320527360000, 12101741169072242688000

Offset: 7

Wolfdieter Lang, Aug 23 2005

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

G. C. Greubel, Table of n, a(n) for n = 7..440

Column 7 of A008297 and unsigned A111596.
Column 6 of A001778.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n-7)*Binomial(n, 7)*Binomial(n-1, 6): n in [7..30]]; // G. C. Greubel, May 10 2021

k = 7; a[n_] := n!*Binomial[n-1, k-1]/k!; Table[a[n], {n, k, 22}]  (* Jean-François Alcover, Jul 09 2013 *)

[factorial(n-7)*binomial(n, 7)*binomial(n-1, 6) for n in (7..30)] # G. C. Greubel, May 10 2021

E.g.f.: ((x/(1-x))^7)/7!.
a(n) = (n!/7!)*binomial(n-1, 7-1).
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} (binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) ) ) then a(n+1) = (-1)^n*f(n,6,-8), (n>=6). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=7} 1/a(n) = 6342*(Ei(1) - gamma) - 8988*e + 80374/5, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=7} (-1)^(n+1)/a(n) = 170142*(gamma - Ei(-1)) - 101640/e - 490714/5, where Ei(-1) = -A099285. (End)

A134141 Generalized unsigned Stirling1 triangle, S1p(7).

Original entry on oeis.org

1, 7, 1, 56, 21, 1, 504, 371, 42, 1, 5040, 6440, 1295, 70, 1, 55440, 114520, 36225, 3325, 105, 1, 665280, 2116800, 983920, 135975, 7105, 147, 1, 8648640, 40884480, 26714800, 5199145, 398860, 13426, 196, 1, 121080960, 826338240, 735469280

Offset: 1

Wolfdieter Lang, Oct 12 2007

Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A092082(n, m) =: S2(7; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m, m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+6 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 05 2007
A triangle of numbers related to triangle A132166.
a(n,1)= A001730(n,5), n>=1. a(n,m)=: S1p(7; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n, m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n, m) (unsigned Lah numbers). S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352, S1p(5; n,m)= A049353(n,m), S1p(6; n,m)= A049374(n, m).
The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			{1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.
a(4,2)= 371 = 4*(7*8)+3*(7*7) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*7*8)=56 colored versions, e.g., ((1c1),(2c1,3c7,4c5)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 7 colors, c1..c7, can be chosen and the vertex labeled 4 with j=2 can come in 8 colors, e.g., c1..c8. Therefore there are 4*((1)*(1*7*8))=224 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*7)*(1*7))=147 such forests, e.g. ((1c1,3c4)(2c1,4c7)) or ((1c1,3c6)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 05 2007

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

First column A001730(n+5), n>=1.

Row sums A132164. Alternating row sums A132165.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+6)!/6!, 9); # Peter Luschny, Jan 27 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (6*m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 39]] (* _Jean-François Alcover, Jun 01 2011, after formula *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(# + 6)!/6! &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+6)/factorial(6), 10) # Peter Luschny, Jan 18 2016

a(n, m) = n!*A132166(n, m)/(m!*6^(n-m)); a(n, m) = (6*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

A134280 Triangle of numbers obtained from the partition array A134279.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 102, 6, 1, 22176, 1452, 102, 6, 1, 576576, 32868, 1668, 102, 6, 1, 17873856, 779328, 35244, 1668, 102, 6, 1, 643458816, 23912064, 843480, 36540, 1668, 102, 6, 1, 26381811456, 812173824, 25416072, 857736, 36540, 1668, 102, 6, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(6)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [6,1]; [66,6,1]; [1056,102,6,1]; [22176,1452,102,6,1]; ...

W. Lang, First 10 rows and more.

Cf. A134275 (S2(5)').
Cf. A134281 (row sums).
Cf. A134282 (alternating row sums).

a(n,m)=sum(product(S2(6;j,1)^e(n,m,k,j),j=1..n),k=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,k,j) is the exponent of j in the k-th m part partition of n. S2(6;j,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5)(quintuple- or 5-factorials).

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A049410 A triangle of numbers related to triangle A049325.

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A059115 Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).

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A134146 Triangle of numbers obtained from the partition array A134145.

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A134275 Triangle of numbers obtained from the partition array A134274.

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A176013 Triangle, read by rows, T(n, k) = (-1)^n * n!/(k*k!) * binomial(n-1, k-1) * binomial(n, k-1).

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Magma

Mathematica

Sage

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A048786 Triangle of coefficients of certain exponential convolution polynomials.

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Maple

Mathematica

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A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

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A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

A176013 Triangle, read by rows, T(n, k) = (-1)^n * n!/(kk!) binomial(n-1, k-1) * binomial(n, k-1).