A008278 -id:A008278 - cp's OEIS frontend

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 7, 10, 15, 52, 15, 20, 27, 37, 52, 203, 52, 67, 87, 114, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

Original entry on oeis.org

1, 3, 13, 76, 571, 5309, 59341, 780149

Offset: 0

Gus Wiseman, Jul 25 2020

A number has distinct prime exponents iff its prime signature is strict.

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(2) = 13 ways:
  12/1/1  12/2/1  12/3/1  12/4/1  12/12/1
          12/2/2  12/3/3  12/4/2  12/12/2
                          12/4/4  12/12/3
                                  12/12/4
                                  12/12/12

A000258 shifted once to the left is dominated by this sequence.
A336422 is the generalization to non-superprimorials.
A000110 counts divisors of superprimorials with distinct prime exponents.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
Cf. A000005, A008278, A027423, A071625, A118914, A124010, A327498, A336417, A336419, A336420, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
strsig[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[chern[n]],d_?strsig:>Count[Divisors[d],e_?strsig]]],{n,0,5}]

A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 7, 1, 24, 60, 50, 15, 1, 120, 360, 390, 180, 31, 1, 720, 2520, 3360, 2100, 602, 63, 1, 5040, 20160, 31920, 25200, 10206, 1932, 127, 1, 40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1, 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1

Offset: 0

Philippe Deléham, Aug 20 2007

Old name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.
Vandervelde (2018) refers to this as the Worpitzky number triangle - N. J. A. Sloane, Mar 27 2018 [Named after the German mathematician Julius Daniel Theodor Worpitzky (1835-1895). - Amiram Eldar, Jun 24 2021]
Triangle given by A123125*A007318 (as infinite lower triangular matrices), A123125 = Euler's triangle, A007318 = Pascal's triangle; A007318*A123125 gives A046802.
Taylor coefficients of Eulerian polynomials centered at 1. - Louis Zulli, Nov 28 2015
A signed refinement is A263634. - Tom Copeland, Nov 14 2016
With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of this entry (the Worpitsky triangle, A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020

			Triangle begins:
1
1      1
2      3       1
6      12      7       1
24     60      50      15      1
120    360     390     180     31      1
720    2520    3360    2100    602     63      1
5040   20160   31920   25200   10206   1932    127    1
40320  181440  332640  317520  166824  46620   6050   255   1
362880 1814400 3780000 4233600 2739240 1020600 204630 18660 511 1
...

G. C. Greubel, Table of n, a(n) for n = 0..1034 [a(438) and a(901) corrected by _Georg Fischer_, Nov 10 2021]
Nguyen-Huu-Bong, Some Combinatorial Properties of Summation Operators, J. Comb. Theory, Ser. A, Vo. 11, No. 3 (1971), pp. 213-221. See Table on page 214.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.
Sam Vandervelde, The Worpitzky Numbers Revisited, Amer. Math. Monthly, Vol. 125, No. 3 (2018), pp. 198-206.

Essentially reverse of A028246.
Variants are A130850, A075263, A163626.
Cf. A000142, A001710, A005460, A005461, A005462, A005463, A005464, A263634.
Cf. A019538, A028246, A046802, A090582, A119879, A123125, A248727.

Table[(n-k)!*StirlingS2[n+1, n-k+1], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Nov 15 2015 *)

t(n, k) = (n-k)!*stirling(n+1, n-k+1, 2);
tabl(nn) = for (n=0, 10, for (k=0, n, print1(t(n,k),", ")); print()); \\ Michel Marcus, Nov 16 2015

from sage.combinat.combinat import eulerian_number
def A130850(n, k):
    return add(eulerian_number(n, j)*binomial(n-j, k) for j in (0..n))
for n in (0..7): [A130850(n, k) for k in (0..n)] # Peter Luschny, May 21 2013

T(n,k) = (-1)^k*A075263(n,k).
T(n,k) = (n-k)!*A008278(n+1,k+1).
T(n,n-1) = 2^n - 1 for n > 0. - Derek Orr, Dec 31 2015
E.g.f.: x/(e^(-x*t)*(1+x)-1). - Tom Copeland, Nov 14 2016
Sum_{k=1..floor(n/2)} T(n,2k) = Sum_{k=0..floor(n/2)} T(n,2k+1) = A000670(n). - Jacob Sprittulla, Oct 03 2021

New name from Peter Luschny, May 21 2013

A106800 Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 0, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1, 0

Offset: 0

N. J. A. Sloane

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 19 2005

			From _Gheorghe Coserea_, Jan 30 2017: (Start)
Triangle starts:
  n\k  [0]  [1]   [2]    [3]    [4]    [5]    [6]   [7] [8] [9]
  [0]   1;
  [1]   1,   0;
  [2]   1,   1,    0;
  [3]   1,   3,    1,     0;
  [4]   1,   6,    7,     1,     0;
  [5]   1,  10,   25,    15,     1,     0;
  [6]   1,  15,   65,    90,    31,     1,     0;
  [7]   1,  21,  140,   350,   301,    63,     1,    0;
  [8]   1,  28,  266,  1050,  1701,   966,   127,    1,  0;
  [9]   1,  36,  462,  2646,  6951,  7770,  3025,  255,  1,  0;
  ...
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, table 2.14.1 at page 24.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
Eric Weisstein's World of Mathematics, Bell Polynomial.

See A008277 and A048993, which are the main entries for this triangle of numbers.
The Stirling1 counterpart is A054654.
Row sum: A000110.
Column 0: A000012.
Column 1: A000217.
Main Diagonal: A000007.
1st minor diagonal: A000012.
2nd minor diagonal: A000225.
3rd minor diagonal: A000392.
Cf. A008278, A340556.

seq(seq(Stirling2(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

Table[ StirlingS2[n, m], {n, 0, 10}, {m, n, 0, -1}]//Flatten (* Robert G. Wilson v, Jan 30 2017 *)

N=11; x='x+O('x^N); t='t; concat(apply(p->Vec(p), Vec(serlaplace(exp(t*(exp(x)-1))))))  \\ Gheorghe Coserea, Jan 30 2017
{T(n, k) = my(A, B); if( n<0 || k>n, 0, A = B = exp(x + x * O(x^n)); for(i=1, n, A = x * A'); polcoeff(A / B, n-k))}; /* Michael Somos, Aug 16 2017 */

flatten([[stirling_number2(n, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 11 2021

A(x;t) = exp(t*(exp(x)-1)) = Sum_{n>=0} P_n(t) * x^n/n!, where P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k). - Gheorghe Coserea, Jan 30 2017
Also, P_n(t) * exp(t) = (t * d/dt)^n exp(t). - Michael Somos, Aug 16 2017
T(n, k) = Sum_{j=0..k} E2(k, j)*binomial(n + k - j, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0

Offset: 0

Gus Wiseman, Aug 03 2020

Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  0
  1  2  1  2  1
  1  3  1  3  2  0
  1  3  2  5  3  3  2  1
  1  4  2  7  4  4  3  2  0
  1  4  2  7  4  5  7  7  6  3  2  0
  1  4  2  8  8  9 10 11 11  7  8  5  2  0
  1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
  1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
  1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
Row n = 7 counts the following divisors:
  1  2  4  8   16  48   144  720   {}
     3  9  12  24  72   360  1008
     5     18  40  80   504
     7     20  56  112
           28
           45
           63

A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

A106338 Triangle T, read by rows, equal to the matrix inverse of the triangle defined by [T^-1](n,k) = A075263(n,k)/n!, for n>=k>=0.

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -9, 14, -6, 1, -45, 110, -90, 24, 1, -585, 1670, -1710, 744, -120, 1, -21105, 61670, -66150, 32424, -7560, 720, 1, -1858185, 5439350, -5864670, 2925384, -728280, 91440, -5040, 1, -367958745, 1077215510, -1161894510, 580489224, -145567800, 18961200, -1285200, 40320, 1

Offset: 0

Paul D. Hanna, May 01 2005

Row sums are zero after the initial row. Absolute row sums equal A106339.

			Triangle begins:
1;
1,-1;
1,-3,2;
1,-9,14,-6;
1,-45,110,-90,24;
1,-585,1670,-1710,744,-120;
1,-21105,61670,-66150,32424,-7560,720;
1,-1858185,5439350,-5864670,2925384,-728280,91440,-5040; ...
The matrix inverse T^-1 begins:
1;
1,1;
1,3/2,1/2;
1,2,7/6,1/6;
1,5/2,25/12,5/8,1/24;
1,3,13/4,3/2,31/120,1/120;
1,7/2,14/3,35/12,301/360,7/80,1/720; ...
where [T^-1](n,k) = A075263(n,k)/n!.
Each row n of the matrix inverse equals the initial
(n+1) fractional coefficients of (x/log(1+x))^n,
which are listed below for n=1,2,3,...,9:
1; 1/2,-1/12,1/24,-19/720,3/160,-863/60480,275/24192,...
1,1; 1/12,0,-1/240,1/240,-221/60480,19/6048,...
1,3/2,1/2; 0,1/240,-1/480,1/945,-11/20160,47/172800,...
1,2,7/6,1/6; -1/720,0,1/3024,-1/3024,199/725760,...
1,5/2,25/12,5/8,1/24; 0,-1/6048,1/12096,-19/725760,...
1,3,13/4,3/2,31/120,1/120; 1/30240,0,-1/57600,1/57600,...
1,7/2,14/3,35/12,301/360,7/80,1/720; 0,1/172800,...
1,4,19/3,5,81/40,23/60,127/5040,1/5040; -1/1209600,0,...
1,9/2,33/4,63/8,331/80,37/32,605/4032,17/2688,1/40320; 0,...

Cf. A075263, A106340, A106339, A008278, A002206.

rows = 10; Tinv = Table[(1/n!)*PadRight[CoefficientList[x^(n+1)*Sum[k^n * (1-x)^k, {k, 0, Infinity}], x], rows], {n, 0, rows-1}]; T = Inverse[Tinv ]; Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 11 2017 *)

T(n,k)=(M=matrix(n+1,n+1,m,j,if(m>=j, polcoeff((-x/log(1-x+x^2*O(x^n)))^m,j-1)))^-1)[n+1,k+1]

T(n,k)=(-1)^n*k!*(matrix(n+1,n+1,r,c,if(r>=c,(r-c)!* sum(m=0,r-c+1,(-1)^(r-c+1-m)*m^r/m!/(r-c+1-m)!)))^-1)[n+1,k+1]

Also, T(n, k) = k!*A106340(n, k), where A106340 is the matrix inverse of the triangle formed from (n-k)!*A008278(n, k), n>=k>=0 and A008278 is the triangle of Stirling numbers of 2nd kind.

cp's OEIS Frontend

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

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A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

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A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

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A106800 Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.

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A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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A106338 Triangle T, read by rows, equal to the matrix inverse of the triangle defined by [T^-1](n,k) = A075263(n,k)/n!, for n>=k>=0.

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A143058 A007318 * [1, 6, 7, 1, 0, 0, 0, ...].

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