A132056 -id:A132056 - cp's OEIS frontend

A132060 Row sums of triangle A132056 (S2(8), Stirling2 generalization).

1, 9, 145, 3361, 101601, 3786601, 167756689, 8611201665, 502522701121, 32854723149961, 2378687990172561, 188913154677765409, 16328988725252964385, 1525987835722540609641, 153312690475268723848081

Offset: 1

Wolfdieter Lang Sep 14 2007

Generalized Bell numbers B(8,1;n).

Cf. A092084 (row sums of S2(7)=A092082).

With[{nn=30},Rest[CoefficientList[Series[Exp[-1+(1-7x)^(-1/7)]-1,{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Sep 23 2011 *)

a(n)=sum(A132056(n, m), m=1..n), n>=1.

E.g.f.: exp(-1+(1-7*x)^(-1/7)) - 1.

A132061 Alternating row sums of triangle A132056 (S2(8), Stirling2 generalization).

Original entry on oeis.org

1, 7, 97, 2015, 55841, 1935719, 80574433, 3915183103, 217530794305, 13603055116679, 945542295992801, 72321915976403807, 6036466379411066977, 545983089637963491175, 53194885608199879974241

Offset: 0

Wolfdieter Lang Sep 14 2007

Cf. A092085 (alternating row sums of S2(7)=A092082).

a(n)= -sum(A132056(n, m)*(-1)^m, m=1..n), n>=1.

E.g.f.:-(exp(1 - (1-7*x)^(-1/7)) - 1).

A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 36, 12, 1, 120, 240, 120, 20, 1, 720, 1800, 1200, 300, 30, 1, 5040, 15120, 12600, 4200, 630, 42, 1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1, 3628800, 16329600

Offset: 1

Miklos Kristof, Apr 25 2005

T(n,k) is the number of partially ordered sets (posets) on n elements that consist entirely of k chains. For example, T(4, 3)=12 since there are exactly 12 posets on {a,b,c,d} that consist entirely of 3 chains. Letting ab denote a<=b and using a slash "/" to separate chains, the 12 posets can be given by a/b/cd, a/b/dc, a/c/bd, a/c/db, a/d/bc, a/d/cb, b/c/ad, b/c/da, b/d/ac, b/d/ca, c/d/ab and c/d/ba, where the listing of the chains is arbitrary (e.g., a/b/cd = a/cd/b =...cd/b/a). - Dennis P. Walsh, Feb 22 2007
Also the matrix product |S1|.S2 of Stirling numbers of both kinds.
This Lah triangle is a lower triangular matrix of the Jabotinsky type. See the column e.g.f. and the D. E. Knuth reference given in A008297. - Wolfdieter Lang, Jun 29 2007
The infinitesimal matrix generator of this matrix is given in A132710. See A111596 for an interpretation in terms of circular binary words and generalized factorials. - Tom Copeland, Nov 22 2007
Three combinatorial interpretations: T(n,k) is (1) the number of ways to split [n] = {1,...,n} into a collection of k nonempty lists ("partitions into sets of lists"), (2) the number of ways to split [n] into an ordered collection of n+1-k nonempty sets that are noncrossing ("partitions into lists of noncrossing sets"), (3) the number of Dyck n-paths with n+1-k peaks labeled 1,2,...,n+1-k in some order. - David Callan, Jul 25 2008
Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = D where D(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 21 2008
An e.g.f. for the row polynomials of A(n,k) = T(n,k)*a(n-k) is exp[a(.)* D_x * x^2] exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,1)*a(.)*x], umbrally, where [(.)! Lag(.,x,1)]^n = n! Lag(n,x,1) is a normalized Laguerre polynomial of order 1. - Tom Copeland, Aug 29 2008
Triangle of coefficients from the Bell polynomial of the second kind for f = 1/(1-x). B(n,k){x1,x2,x3,...} = B(n,k){1/(1-x)^2,...,(j-1)!/(1-x)^j,...} = T(n,k)/(1-x)^(n+k). - Vladimir Kruchinin, Mar 04 2011
The triangle, with the row and column offset taken as 0, is the generalized Riordan array (exp(x), x) with respect to the sequence n!*(n+1)! as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!^2 is A021009 unsigned). - Peter Bala, Aug 15 2013
For a relation to loop integrals in QCD, see p. 33 of Gopakumar and Gross and Blaizot and Nowak. - Tom Copeland, Jan 18 2016
Also the Bell transform of (n+1)!. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
Also the number of k-dimensional flats of the n-dimensional Shi arrangement. - Shuhei Tsujie, Apr 26 2019
The numbers T(n,k) appear as coefficients when expanding the rising factorials (x)^k = x(x+1)...(x+k-1) in the basis of falling factorials (x)k = x(x-1)...(x-k+1). Specifically, (x)^n = Sum{k=1..n} T(n,k) (x)k. - _Jeremy L. Martin, Apr 21 2021

			T(1,1) = C(1,1)*0!/0! = 1,
T(2,1) = C(2,1)*1!/0! = 2,
T(2,2) = C(2,2)*1!/1! = 1,
T(3,1) = C(3,1)*2!/0! = 6,
T(3,2) = C(3,2)*2!/1! = 6,
T(3,3) = C(3,3)*2!/2! = 1,
Sheffer a-sequence recurrence: T(6,2)= 1800 = (6/3)*120 + 6*240.
B(n,k) =
   1/(1-x)^2;
   2/(1-x)^3,  1/(1-x)^4;
   6/(1-x)^4,  6/(1-x)^5,  1/(1-x)^6;
  24/(1-x)^5, 36/(1-x)^6, 12/(1-x)^7, 1/(1-x)^8;
The triangle T(n,k) begins:
  n\k      1       2       3      4      5     6    7  8  9 ...
  1:       1
  2:       2       1
  3:       6       6       1
  4:      24      36      12      1
  5:     120     240     120     20      1
  6:     720    1800    1200    300     30     1
  7:    5040   15120   12600   4200    630    42    1
  8:   40320  141120  141120  58800  11760  1176   56  1
  9:  362880 1451520 1693440 846720 211680 28224 2016 72  1
  ...
Row n=10: [3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1]. - _Wolfdieter Lang_, Feb 01 2013
From _Peter Bala_, Feb 24 2025: (Start)
The array factorizes as an infinite product (read from right to left):
  /  1                \        /1             \^m /1           \^m /1           \^m
  |  2    1            |      | 0   1          |  |0  1         |  |1  1         |
  |  6    6   1        | = ...| 0   0   1      |  |0  1  1      |  |0  2  1      |
  | 24   36  12   1    |      | 0   0   1  1   |  |0  0  2  1   |  |0  0  3  1   |
  |120  240 120  20   1|      | 0   0   0  2  1|  |0  0  0  3  1|  |0  0  0  4  1|
  |...                 |      |...             |  |...          |  |...          |
where m = 2. Cf. A008277 (m = 1), A035342 (m = 3), A035469 (m = 4), A049029 (m = 5) A049385 (m = 6), A092082 (m = 7), A132056 (m = 8), A223511 - A223522 (m = 9 through 20), A001497 (m = -1), A004747 (m = -2), A000369 (m = -3), A011801 (m = -4), A013988 (m = -5). (End)

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Peter Bala, Factorising (r,b)-Stirling arrays
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5
Jean-Paul Blaizot and Maciej A. Nowak, Large N_c confinement and turbulence, arXiv:0801.1859 [hep-th], 2008.
David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008.
Pietro Codara, Ottavio M. D'Antona, and Pavol Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
Tom Copeland, Mathemagical Forests, Addendum to Mathemagical Forests, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions, A Class of Differential Operators and the Stirling Numbers
Siad Daboul, Jan Mangaldan, Michael Z. Spivey and Peter Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., 86 (2013), 39-47.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 33.
G. H. E. Duchamp et al., Feynman graphs and related Hopf algebras, J. Phys. (Conf Ser) 30 (2006) 107-118.
Rajesh Gopakumar and David J. Gross, Mastering the master field, arXiv:hep-th/9411021, 1994.
Gábor Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4. - From _Tom Copeland_, Oct 01 2015
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Donald E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
MacTutor History of Mathematics archive: Biography of Ivo Lah.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019. See p. 18.
Michael Penn, Lah Numbers and an appearance of exponential generating functions, YouTube video, 2025.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019. See p. 4.
Tilman Piesk, Illustration of the first four rows
Kornelia Ufniarz and Grzegorz Siudem, Combinatorial origins of the canonical ensemble, arXiv:2008.00244 [math-ph], 2020. See p. 5.
Weiping Wang and Tianming Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Wikipedia, Lah number

Cf. A000262, A103194, A105220.
Triangle of Lah numbers (A008297) unsigned.
Cf. A111596 (signed triangle with extra n=0 row and m=0 column).
Cf. A130561 (for a natural refinement).
Cf. A094638 (for differential operator representation).
Cf. A248045 (central terms), A002868 (row maxima).
Cf, A059110.
Cf. A001263, A008277.
Cf. A089231 (triangle with mirrored rows).
Cf. A271703 (triangle with extra n=0 row and m=0 column).

Flat(List([1..10],n->List([1..n],k->Binomial(n,k)*Factorial(n-1)/Factorial(k-1)))); # Muniru A Asiru, Jul 25 2018

a105278 n k = a105278_tabl !! (n-1) !! (k-1)
a105278_row n = a105278_tabl !! (n-1)
a105278_tabl = [1] : f [1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014, Mar 18 2013

/* As triangle */ [[Binomial(n,k)*Factorial(n-1)/Factorial(k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 31 2014

The triangle: for n from 1 to 13 do seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n) od;
the sequence: seq(seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n),n=1..13);
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> (n+1)!, 9); # Peter Luschny, Jan 27 2016

nn = 9; a = x/(1 - x); f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y a], {x, 0, nn}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 11 2011 *)
nn = 9; Flatten[Table[(j - k)! Binomial[j, k] Binomial[j - 1, k - 1], {j, nn}, {k, j}]] (* Jan Mangaldan, Mar 15 2013 *)
rows = 10;
t = Range[rows]!;
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
T[n_, n_] := 1; T[n_, k_] /;0Oliver Seipel, Dec 06 2024 *)

use ntheory ":all"; say join ", ", map { my $n=$; map { stirling($n,$,3) } 1..$n; } 1..9; # Dana Jacobsen, Mar 16 2017

T(n,k) = Sum_{m=n..k} |S1(n,m)|*S2(m,k), k>=n>=1, with Stirling triangles S2(n,m):=A048993 and S1(n,m):=A048994.
T(n,k) = C(n,k)*(n-1)!/(k-1)!.
Sum_{k=1..n} T(n,k) = A000262(n).
n*Sum_{k=1..n} T(n,k) = A103194(n) = Sum_{k=1..n} T(n,k)*k^2.
E.g.f. column k: (x^(k-1)/(1-x)^(k+1))/(k-1)!, k>=1.
Recurrence from Sheffer (here Jabotinsky) a-sequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n-1,m-1) + n*T(n-1,m). - Wolfdieter Lang, Jun 29 2007
The e.g.f. is, umbrally, exp[(.)!* L(.,-t,1)*x] = exp[t*x/(1-x)]/(1-x)^2 where L(n,t,1) = Sum_{k=0..n} T(n+1,k+1)*(-t)^k = Sum_{k=0..n} binomial(n+1,k+1)* (-t)^k / k! is the associated Laguerre polynomial of order 1. - Tom Copeland, Nov 17 2007
For this Lah triangle, the n-th row polynomial is given umbrally by
  n! C(B.(x)+1+n,n) = (-1)^n C(-B.(x)-2,n), where C(x,n)=x!/(n!(x-n)!),
  the binomial coefficient, and B_n(x)= exp(-x)(xd/dx)^n exp(x), the n-th Bell / Touchard / exponential polynomial (cf. A008277). E.g.,
  2! C(-B.(-x)-2,2) = (-B.(x)-2)(-B.(x)-3) = B_2(x) + 5*B_1(x) + 6 = 6 + 6x + x^2.
  n! C(B.(x)+1+n,n) = n! e^(-x) Sum_{j>=0} C(j+1+n,n)x^j/j! is a corresponding Dobinski relation. See the Copeland link for the relation to inverse Mellin transform. - Tom Copeland, Nov 21 2011
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A008277 (D = (1+x)*d/dx), A035342 (D = (1+x)^3*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). - Peter Bala, Nov 25 2011
T(n,k) = Sum_{i=k..n} A130534(n-1,i-1)*A008277(i,k). - Reinhard Zumkeller, Mar 18 2013
Let E(x) = Sum_{n >= 0} x^n/(n!*(n+1)!). Then a generating function is exp(t)*E(x*t) = 1 + (2 + x)*t + (6 + 6*x + x^2)*t^2/(2!*3!) + (24 + 36*x + 12*x^2 + x^3)*t^3/(3!*4!) + ... . - Peter Bala, Aug 15 2013
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of A059110; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 23 2018
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates the Narayana matrix A001263. - Tom Copeland, Sep 23 2020
T(n,k) = A089231(n,n-k). - Ron L.J. van den Burg, Dec 12 2021
T(n,k) = T(n-1,k-1) + (n+k-1)*T(n-1,k). - Bérénice Delcroix-Oger, Jun 25 2025

Stirling comments and e.g.f.s from Wolfdieter Lang, Apr 11 2007

A045754 7-fold factorials: a(n) = Product_{k=0..n-1} (7*k+1).

Original entry on oeis.org

1, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 337767408000, 21617114112000, 1534815101952000, 119715577952256000, 10175824125941760000, 936175819586641920000, 92681406139077550080000, 9824229050742220308480000, 1110137882733870894858240000

Offset: 0

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 0..338
Index entries for sequences related to factorial numbers.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A008542 (and A085158), A045755.
See also A113134.
Unsigned row sums of triangle A051186 (scaled Stirling1).
First column of triangle A132056 (S2(8)).
Cf. A114799, A084947, A144739, A144827, A147585, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+1) ); # G. C. Greubel, Aug 21 2019

[1] cat [&*[7*j+1: j in [0..n-1]]: n in [1..20]]; // G. C. Greubel, Aug 21 2019

f := n->product( (7*k+1), k=0..(n-1));
G(x):=(1-7*x)^(-1/7): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

FoldList[Times, 1, 7Range[0, 20] + 1] (* Harvey P. Dale, Jan 21 2013 *)

PARI
```
a(n)=prod(k=0,n-1,7*k+1)
```

[7^n*rising_factorial(1/7, n) for n in (0..20)] # G. C. Greubel, Aug 21 2019

a(n) = Sum_{k=0..n} (-7)^(n-k)*A048994(n, k), where A048994 = Stirling-1 numbers.
E.g.f.: (1-7*x)^(-1/7).
G.f.: 1/(1-x/(1-7*x/(1-8*x/(1-14*x/(1-15*x/(1-21*x/(1-22*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-6)^n*Sum_{k=0..n} (7/6)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/G(0), where G(k)= 1 - x*(7*k+1)/(1 - x*(7*k+7)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k+1)/(x*(7*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
a(n) = 7^n * Gamma(n + 1/7) / Gamma(1/7). - Artur Jasinski, Aug 23 2016
a(n) = A114799(7n-6). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-7*n+6)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/7^6)^(1/7)*(Gamma(1/7) - Gamma(1/7, 1/7)). - Amiram Eldar, Dec 19 2022

Additional comments from Philippe Deléham and Paul D. Hanna, Oct 29 2005
Edited by N. J. A. Sloane, Oct 16 2008 at the suggestion of M. F. Hasler, Oct 14 2008
Corrected by Zerinvary Lajos, Apr 03 2009

A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)(x^(1/2)d/dx)^n when n is odd, and of 2^(n/2)(x^(1/2)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 12, 4, 15, 20, 4, 15, 90, 60, 8, 105, 210, 84, 8, 105, 840, 840, 224, 16, 945, 2520, 1512, 288, 16, 945, 9450, 12600, 5040, 720, 32, 10395, 34650, 27720, 7920, 880, 32, 10395, 124740, 207900, 110880, 23760, 2112, 64, 135135, 540540, 540540, 205920, 34320, 2496, 64

Offset: 0

Udita Katugampola, Mar 17 2013

Also coefficients in the expansion of k-th derivative of exp(n*x^2), see Mathematica program. - Vaclav Kotesovec, Jul 16 2013

			Triangle begins:
       1;
       1,      2;
       3,      2;
       3,     12,      4;
      15,     20,      4;
      15,     90,     60,      8;
     105,    210,     84,      8;
     105,    840,    840,    224,    16;
     945,   2520,   1512,    288,    16;
     945,   9450,  12600,   5040,   720,   32;
   10395,  34650,  27720,   7920,   880,   32;
   10395, 124740, 207900, 110880, 23760, 2112, 64;
  135135, 540540, 540540, 205920, 34320, 2496, 64;
  .
Expansion takes the form:
2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx.
2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3.
2^2 (x^(1/2)*d/dx)^4 = 3*d^2/dx^2 + 12*x*d^3/dx^3 + 4*x^2*d^4/dx^4.
2^2 (x^(1/2)*d/dx)^5 = 15*x^(1/2)*d^3/dx^3 + 20*x^(3/2)*d^4/dx^4 + 4*x^(5/2)*d^5/dx^5.
`
`

Vincenzo Librandi, Rows n = 0..60, flattened
U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229 [math.CA], 2014.

Odd rows includes absolute values of A098503 from right to left.

Cf. A223169-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x);
for i from 1 to 13 do
a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
end do;

Flatten[CoefficientList[Expand[FullSimplify[Table[D[E^(n*x^2),{x,k}]/(E^(n*x^2)*(2*n)^Floor[(k+1)/2]),{k,1,13}]]]/.x->1,n]] (* Vaclav Kotesovec, Jul 16 2013 *)

A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)(x^(1/6)d/dx)^n when n is odd, and of 6^(n/2)(x^(5/6)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 6, 7, 6, 7, 84, 36, 91, 156, 36, 91, 1638, 1404, 216, 1729, 4446, 2052, 216, 1729, 41496, 53352, 16416, 1296, 43225, 148200, 102600, 21600, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 5742750, 5301000, 1674000, 200880, 7776

Offset: 0

Udita Katugampola, Mar 20 2013

			Triangle begins:
        1;
        1,        6;
        7,        6;
        7,       84,        36;
       91,      156,        36;
       91,     1638,      1404,      216;
     1729,     4446,      2052,      216;
     1729,    41496,     53352,    16416,     1296;
    43225,   148200,    102600,    21600,     1296;
    43225,  1296750,   2223000,  1026000,   162000,    7776;
  1339975,  5742750,   5301000,  1674000,   200880,    7776;
  1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229 [math.CA], 2014.

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do;

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 6, 7, 84, 36, 91, 1638, 1404, 216, 1729, 41496, 53352, 16416, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656, 49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512

Offset: 1

Udita Katugampola, Mar 23 2013

			Triangle begins:
1;
1, 6;
7, 84, 36;
91, 1638, 1404, 216;
1729, 41496, 53352, 16416, 1296;
43225, 1296750, 2223000, 1026000, 162000, 7776;
1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;
49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512, 12083904, 279936;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Even rows of A223172.

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 9, 1, 153, 27, 1, 3825, 855, 54, 1, 126225, 32895, 2745, 90, 1, 5175225, 1507815, 150930, 6705, 135, 1, 253586025, 80565975, 9205245, 499590, 13860, 189, 1, 14454403425, 4926412575, 623675430, 39180645, 1345050, 25578, 252, 1

Offset: 1

Udita Katugampola, Mar 23 2013

Also the Bell transform of A045755(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			1;
9,1;
153,27,1;
3825,855,54,1;
126225,32895,2745,90,1;
5175225,1507815,150930,6705,135,1;
253586025,80565975,9205245,499590,13860,189,1;
14454403425,4926412575,623675430,39180645,1345050,25578,252,1;

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223512-A223522, A223168-A223172, A223523-A223532.

b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^9*diff(b[j-1],x$1);
end do;
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(8*k+1, k=0..n), 10); # Peter Luschny, Jan 29 2016

rows = 8;
t = Table[Product[8k+1, {k, 0, n}], {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 *)

A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 20, 1, 780, 60, 1, 45240, 4320, 120, 1, 3483480, 382200, 13800, 200, 1, 334414080, 40556880, 1734600, 33600, 300, 1, 38457619200, 5039012160, 243505080, 5699400, 69300, 420, 1

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
20,1;
780,60,1;
45240,4320,120,1;
3483480,382200,13800,200,1;
334414080,40556880,1734600,33600,300,1;
38457619200,5039012160,243505080,5699400,69300,420,1;
5153320972800,718724260800,38155703040,1024322880,15262800,127680,560,1;

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^20*diff(b[j-1],x$1);
end do;

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n is odd, and of 3^(n/2)(x^(2/3)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 3, 4, 3, 4, 24, 9, 28, 42, 9, 28, 252, 189, 27, 280, 630, 270, 27, 280, 3360, 3780, 1080, 81, 3640, 10920, 7020, 1404, 81, 3640, 54600, 81900, 35100, 5265, 243, 58240, 218400, 187200, 56160, 6480, 243, 58240, 1048320, 1965600

Offset: 0

Udita Katugampola, Mar 18 2013

			Triangle begins:
1;
1, 3;
4, 3;
4, 24, 9;
28, 42, 9;
28, 252, 189, 27;
280, 630, 270, 27;
280, 3360, 3780, 1080, 81;
3640, 10920, 7020, 1404, 81;
3640, 54600, 81900, 35100, 5265, 243,
58240, 218400, 187200, 56160, 6480, 243

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do;

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A132061 Alternating row sums of triangle A132056 (S2(8), Stirling2 generalization).

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A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)*(x^(1/6)*d/dx)^n when n is odd, and of 6^(n/2)*(x^(5/6)*d/dx)^n when n is even.

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A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...

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A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)(x^(1/2)d/dx)^n when n is odd, and of 2^(n/2)(x^(1/2)d/dx)^n when n is even.

A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)(x^(1/6)d/dx)^n when n is odd, and of 6^(n/2)(x^(5/6)d/dx)^n when n is even.

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n is odd, and of 3^(n/2)(x^(2/3)d/dx)^n when n is even.