A039692 -id:A039692 - cp's OEIS frontend

A049352 A triangle of numbers related to triangle A030524.

1, 4, 1, 20, 12, 1, 120, 128, 24, 1, 840, 1400, 440, 40, 1, 6720, 16240, 7560, 1120, 60, 1, 60480, 201600, 129640, 27720, 2380, 84, 1, 604800, 2681280, 2275840, 656320, 80080, 4480, 112, 1, 6652800, 38142720, 41370560, 15402240, 2498160, 196560

Offset: 1

Wolfdieter Lang

a(n,1) = A001715(n+2). a(n,m)=: S1p(4; 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).
The signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035469(n,m) := S2(4; 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+3 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001715. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle starts:
{1};
{4,1};
{20,12,1};
{120,128,24,1};
{840,1400,440,40,1};
...
E.g. Row polynomial E(3,x)= 20*x + 12*x^2 + x^3.
a(4,2)=128=4*(4*5)+3*(4*4) 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*4*5)=20 colored versions, e.g. ((1c1),(2c1,3c4,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 4 colors, c1..c4, can be chosen and the vertex labeled 4 with j=2 can come in 5 colors, e.g. c1..c5. Therefore there are 4*((1)*(1*4*5))=80 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*4)*(1*4))=48 such forests, e.g. ((1c1,3c2)(2c1,4c4)) or ((1c1,3c3)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Wolfdieter Lang, First ten rows.

Cf. A049377 (row sums).

Alternating row sums A134137.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+3)!/6, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] := (n!* Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n+3*j-1, 3*j-1], {j, 1, k}])/(3^k*k!); Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(# + 3)!/6&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1),j,1,k))/(3^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n, m) = n!*A030524(n, m)/(m!*3^(n-m)); a(n, m) = (3*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1)))/(3^k*k!). [Vladimir Kruchinin, Apr 01 2011]

A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 0, 3, 6, 1, 0, 0, 15, 10, 1, 0, 0, 15, 45, 15, 1, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0, 0, 0, 0, 10395, 62370

Offset: 1

Wolfdieter Lang

a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; 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))*A001497(n-1,m-1) (signed Bessel triangle). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - Paul Barry, Jan 15 2009

			Triangle a(n,m) (with rows n >= 1 and columns m >= 1) begins as follows:
  1;                 with row polynomial E(1,x) = x;
  1, 1;              with row polynomial E(2,x) = x^2 + x;
  0, 3,  1;          with row polynomial E(3,x) = 3*x^2 + x^3;
  0, 3,  6,   1;     with row polynomial E(4,x) = 3*x^2 + 6*x^3 + x^4;
  0, 0, 15,  10,   1;
  0, 0, 15,  45,  15,   1;
  0, 0,  0, 105, 105,  21,  1;
  0, 0,  0, 105, 420, 210, 28, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows of the array and more.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.

Cf. A000085 (row sums), A001497, A001498, A008275, A008279, A008297, A019590, A030528, A039692.

Variations of this array: A096713, A104556, A122848, A130757.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 28 2016

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
(* Second program: *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[If[#<2, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)) for n >= m >= 1.
a(n, m) = (2*m-n+1)*a(n-1, m) + a(n-1, m-1) for n >= m >= 1 with a(n, m) = 0 for n < m, a(n, 0) := 0, and a(1, 1) = 1. [The 0th column does not appear in this array. - Petros Hadjicostas, Oct 28 2019]
E.g.f. for the m-th column: (x*(1 + x/2))^m/m!.
a(n,m) = A122848(n,m). - R. J. Mathar, Jan 14 2011

A049353 A triangle of numbers related to triangle A030526.

Original entry on oeis.org

1, 5, 1, 30, 15, 1, 210, 195, 30, 1, 1680, 2550, 675, 50, 1, 15120, 34830, 14025, 1725, 75, 1, 151200, 502740, 287280, 51975, 3675, 105, 1, 1663200, 7692300, 5961060, 1482705, 151200, 6930, 140, 1, 19958400, 124740000, 126913500, 41545980

Offset: 1

Wolfdieter Lang

a(n,1)= A001720(n+3). a(n,m)=: S1p(5; 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(n,m).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049029(n,m) := S2(5; 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+4 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001720. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
  {1};
  {5,1};
  {30,15,1}; E.g., row polynomial E(3,x)=30*x+15*x^2+x^3.
  {210,195,30,1};
  ...
a(4,2)= 195 =4*(5*6)+3*(5*5) 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*5*6)=30 colored versions, e.g., ((1c1),(2c1,3c5,4c6)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 5 colors, c1..c5, can be chosen and the vertex labeled 4 with j=2 can come in 6 colors, e.g., c1..c6. Therefore there are 4*((1)*(1*5*6))=120 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*5)*(1*5))=75 such forests, e.g., ((1c1,3c4)(2c1,4c5)) or ((1c1,3c5)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

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

Cf. A049378 (row sums).

Cf. A134139 (alternating row sums).

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+4)!/24, 10); # Peter Luschny, Jan 28 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (4m + 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}]] (* _Jean-François Alcover, Jul 22 2011 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+4)!/24&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+4*j-1,4*j-1),j,1,k))/(4^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n, m) = n!*A030526(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

a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+4*j-1,4*j-1)))/(4^k*k!). - Vladimir Kruchinin, Apr 01 2011

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

Original entry on oeis.org

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1.

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).

Cf. A075499, A075501.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[5^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (5^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.

A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

Original entry on oeis.org

1, 7, 1, 49, 21, 1, 343, 343, 42, 1, 2401, 5145, 1225, 70, 1, 16807, 74431, 30870, 3185, 105, 1, 117649, 1058841, 722701, 120050, 6860, 147, 1, 823543, 14941423, 16235562, 4084101, 360150, 13034, 196, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(7*z) - 1)*x/7) - 1.

			[1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      7        1
*     49       21        1
*    343      343       42       1
*   2401     5145     1225      70      1
*  16807    74431    30870    3185    105     1
* 117649  1058841   722701  120050   6860   147   1
* 823543 14941423 16235562 4084101 360150 13034 196 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.

Cf. A075501, A075503.

Flatten[Table[7^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (7^(n-m)) * stirling2(n, m).
a(n, m) = 7*m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*7)^(n-m))/(m-1)! for n >= m >= 1, else 0.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.
E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.

A075504 Stirling2 triangle with scaled diagonals (powers of 9).

Original entry on oeis.org

1, 9, 1, 81, 27, 1, 729, 567, 54, 1, 6561, 10935, 2025, 90, 1, 59049, 203391, 65610, 5265, 135, 1, 531441, 3720087, 1974861, 255150, 11340, 189, 1, 4782969, 67493007, 57041334, 11160261, 765450, 21546, 252, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(9*z) - 1)*x/9) - 1.
Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.

			[1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       9        1
*      81       27        1
*     729      567       54        1
*    6561    10935     2025       90      1
*   59049   203391    65610     5265    135     1
*  531441  3720087  1974861   255150  11340   189   1
* 4782969 67493007 57041334 11160261 765450 21546 252 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A075503, A075505.

Columns 2-7 are A076008-A076013.

Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (9^(n-m)) * stirling2(n, m).
a(n, m) = Sum_{p=0..m-1} (A075513(m, p)*((p+1)*9)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 9m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-9k*x), m >= 1.
E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.

A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 0, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 0

Offset: 0

Wolfdieter Lang Sep 14 2007

This is a Jabotinsky type exponential convolution triangle related to A001147 (double factorials). For Jabotinsky type triangles See the D. E. Knuth reference given under A039692.
The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).
For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.
This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).
This triangle (or the A001497 subtriangle) appears as generalized Stirling numbers of the second kind, S2p(-1,n,m):=S2(-k;m,m)*(-1)^(n-m) for k=1, eqs. (27)-(29) of the W. Lang reference.
Also the Bell transform of the double factorial of odd numbers A001147. For the Bell transform of the double factorial of even numbers A000165 see A039683. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015

			[1]
[0,      1]
[0,      1,      1]
[0,      3,      3,     1]
[0,     15,     15,     6,     1]
[0,    105,    105,    45,    10,    1]
[0,    945,    945,   420,   105,   15,   1]
[0,  10395,  10395,  4725,  1260,  210,  21,  1]
[0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]

Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012
Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]

Leonard Carlitz, A Note on the Bessel Polynomials, Duke Math. J. 24 (2) (1957), 151-162. [_Emanuele Munarini_, Oct 10 2017]
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.

Columns m=1: A001147.
Row sums give [1, A001515]. Alternating row sums give [1, -A000806].
Cf. A122850. - R. J. Mathar, Mar 20 2009
Cf. A039683, A264428.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
# Alternative:
egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12):
coefx := n -> n!*coeff(serx, x, n): row := n -> seq(coeff(coefx(n), y, k), k = 0..n): for n from 0 to 8 do row(n) od;  # Peter Luschny, Apr 25 2024

Table[If[k <= n, Binomial[2n-2k,n-k] Binomial[2n-k-1,k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(2#-1)!!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

# uses[bell_transform from A264428]
def A132062_row(n):
    a = sloane.A001147
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A132062_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

a(n,m)=0 if n

E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).

From Emanuele Munarini, Oct 10 2017: (Start)

a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).

The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)

T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - Peter Luschny, Apr 25 2024

A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -8, 0, 1, 24, 0, -20, 0, 1, 0, 184, 0, -40, 0, 1, -720, 0, 784, 0, -70, 0, 1, 0, -8448, 0, 2464, 0, -112, 0, 1, 40320, 0, -52352, 0, 6384, 0, -168, 0, 1, 0, 648576, 0, -229760, 0, 14448, 0, -240, 0, 1, -3628800, 0, 5360256, 0, -804320, 0, 29568, 0, -330, 0, 1

Offset: 1

N. J. A. Sloane

|T(n,k)| gives the sum of the M_2 multinomial numbers (A036039) for those partitions of n with exactly k odd parts. E.g.: |T(6,2)| = 144 + 40 = 184 from the partitions of 6 with exactly two odd parts, namely (1,5) and (3,3), with M_2 numbers 144 and 40. Proof via the general Jabotinsky triangle formula for |T(n,k)| using partitions of n into k parts and their M_3 numbers (A036040). Then with the special e.g.f. of the (unsigned) k=1 column, f(x):= arctanh(x), only odd parts survive and the M_3 numbers are changed into the M_2 numbers. For the Knuth reference on Jabotinsky triangles see A039692. - Wolfdieter Lang, Feb 24 2005 [The first two sentences have been corrected thanks to the comment by José H. Nieto S. given below. - Wolfdieter Lang, Jan 16 2012]
|T(n,k)| gives the number of permutations of {1,2,...,n} (degree n permutations) with the number of odd cycles equal to k. E.g.: |T(5,3)|= 20 from the 20 degree 5 permutations with cycle structure (.)(.)(...). Proof: Use the cycle index polynomial for the symmetric group S_n (see the M_2 array A036039 or A102189) together with the partition interpretation of |T(n,k)| given above. - Wolfdieter Lang, Feb 24 2005 [See the following José H. Nieto S. correction. - Wolfdieter Lang, Jan 16 2012]
The first sentence of the above comment is inexact, it should be "|T(n,k)| gives the number of degree n permutations which decompose into exactly k odd cycles". The number of degree n permutations with k odd cycles (and, possibly, other cycles of even length) is given by A060524. - José H. Nieto S., Jan 15 2012
The unsigned triangle with e.g.f. exp(x*arctanh(z)) is the associated Jabotinsky type triangle for the Sheffer type triangle A060524. See the comments there. - Wolfdieter Lang, Feb 24 2005
Also the Bell transform of the sequence (-1)^(n/2)*A005359(n) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
   1;
   0,   1;
  -2,   0,   1;
   0,  -8,   0,   1;
  24,   0, -20,   0,   1;
   0, 184,   0, -40,   0,   1;
  ...
O.g.f. for fifth subdiagonal: (24*t+16*t^2)/(1-t)^7 = 24*t + 184*t^2 + 784*t^3 + 2404*t^4 + ....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial

Essentially same as A008309, which is the main entry for this sequence.

Row sums (unsigned) give A000246(n); signed row sums give A002019(n), n>=1. A137513.

A049218 := proc(n,k)(-1)^((3*n+k)/2) *add(2^(j-k)*n!/j! *stirling1(j,k) *binomial(n-1,j-1),j=k..n) ; end proc: # R. J. Mathar, Feb 14 2011
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::odd, 0, (-1)^(n/2)*n!), 10); # Peter Luschny, Jan 28 2016

t[n_, k_] := (-1)^((3n+k)/2)*Sum[ 2^(j-k)*n!/j!*StirlingS1[j, k]*Binomial[n-1, j-1], {j, k, n}]; Flatten[ Table[ t[n, k], {n, 1, 11}, {k, 1, n}]] (* Jean-François Alcover, Dec 06 2011, after Vladimir Kruchinin *)
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[If[OddQ[#], 0, (-1)^(#/2)*#!]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

T(n,k)=polcoeff(serlaplace(atan(x)^k/k!), n)

E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(n, k) x^n/n!.
T(n,k) = ((-1)^((3*n+k)/2)*n!/2^k)*Sum_{i=k..n} 2^i*binomial(n-1,i-1)*Stirling1(i,k)/i!. - Vladimir Kruchinin, Feb 11 2011
E.g.f.: exp(t*arctan(x)) = 1 + t*x + t^2*x^2/2! + t*(t^2-2)*x^3/3! + .... The unsigned row polynomials are the Mittag-Leffler polynomials M(n,t/2). See A137513. The compositional inverse (with respect to x) (x-t/2*log((1+x)/(1-x)))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3!+ (24*t+16*t^2)/(1-t)^7*x^5/5! + .... The rational functions in t generate the (unsigned) diagonals of the table. See the Bala link. - Peter Bala, Dec 04 2011

Additional comments from Michael Somos

A049374 A triangle of numbers related to triangle A030527.

Original entry on oeis.org

1, 6, 1, 42, 18, 1, 336, 276, 36, 1, 3024, 4200, 960, 60, 1, 30240, 66024, 23400, 2460, 90, 1, 332640, 1086624, 557424, 87360, 5250, 126, 1, 3991680, 18805248, 13349952, 2916144, 255360, 9912, 168, 1, 51891840, 342486144, 325854144, 95001984

Offset: 1

Wolfdieter Lang

a(n,1) = A001725(n+4). a(n,m)=: S1p(6; 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).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049385(n,m) =: S2(6; n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} (a(n,m)*x^m), 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+5 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007

			Triangle begins
       1;
       6,       1;
      42,      18,      1;
     336,     276,     36,     1;
    3024,    4200,    960,    60,    1;
   30240,   66024,  23400,  2460,   90,   1;
  332640, 1086624, 557424, 87360, 5250, 126, 1;
E.g., row polynomial E(3,x) = 42*x + 18*x^2 + x^3.
a(4,2) = 276 = 4*(6*7) + 3*(6*6) 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*6*7)=42 colored versions, e.g., ((1c1),(2c1,3c6,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 6 colors, c1..c6, can be chosen and the vertex labeled 4 with j=2 can come in 7 colors, e.g., c1..c7. Therefore there are 4*((1)*(1*6*7))=168 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*6)*(1*6))=108 such forests, e.g., ((1c1,3c4)(2c1,4c6)) or ((1c1,3c5)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Muniru A Asiru, Table of n, a(n) for n = 1..1275
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A049402 (row sums), A134140 (alternating row sums).

Flat(List([1..10],n->Factorial(n)*List([1..n],k->Sum([1..k],j->(-1)^(k-j)*Binomial(k,j)*Binomial(n+5*j-1,5*j-1)/(5^k*Factorial(k)))))); # Muniru A Asiru, Jun 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+5)!/120, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] = n!*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n + 5j - 1, 5j - 1]/(5^k*k!), {j, 1, k}] ;
Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ][[1 ;; 40]]
(* Jean-François Alcover, Jun 01 2011, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+5)!/120&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k)=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1),j,1,k))/(5^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n,k)=(n!*sum(j=1,k,(-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1)))/(5^k*k!);
for(n=1,12,for(k=1,n,print1(a(n,k),", "));print()); /* print triangle */ /* Joerg Arndt, Apr 01 2011 */

a(n, m) = n!*A030527(n, m)/(m!*5^(n-m)); a(n, m) = (5*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n < m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(5 - 10*x + 10*x^2 - 5*x^3 + x^4)/(5*(1-x)^5))^m)/m!.

a(n,k) = n!* Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1) /(5^k*k!). - Vladimir Kruchinin, Apr 01 2011

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A051142 Generalized Stirling number triangle of first kind.

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A049352 A triangle of numbers related to triangle A030524.

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A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

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A049353 A triangle of numbers related to triangle A030526.

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A075500 Stirling2 triangle with scaled diagonals (powers of 5).

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A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

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A075504 Stirling2 triangle with scaled diagonals (powers of 9).

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