A130191 Square of the Stirling2 matrix A048993.

1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 15, 32, 12, 1, 0, 52, 175, 110, 20, 1, 0, 203, 1012, 945, 280, 30, 1, 0, 877, 6230, 8092, 3465, 595, 42, 1, 0, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 0, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1

Offset: 0

Author

Wolfdieter Lang, Jun 01 2007

Keywords

• nonn
• tabl
• easy

Comments

Without row n=0 and column k=0 this is triangle A039810.

This is an associated Sheffer matrix with e.g.f. of the m-th column ((exp(f(x))-1)^m)/m! with f(x)=:exp(x)-1.

The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - Peter Luschny, Apr 19 2015

Also the Bell transform of shifted Bell numbers A000110(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Examples

			Triangle starts:
  1;
  0,   1;
  0,   2,    1;
  0,   5,    6,    1;
  0,  15,   32,   12,    1;
  0,  52,  175,  110,   20,   1;
  0, 203, 1012,  945,  280,  30,  1;
  0, 877, 6230, 8092, 3465, 595, 42, 1;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
• Wolfdieter Lang, First 10 rows and more
• John Riordan, Letter, Apr 28 1976. (See third page)

Crossrefs

Columns k=0..3 give A000007, A000110 (for n > 0), A000558, A000559.

Row sums: A000258.

Alternating row sums: A130410.

T(2n,n) gives A321712.

Cf. A039810 (another version), A048993.

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> combinat:-bell(n+1), 9); # Peter Luschny, Jan 27 2016

Mathematica

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[BellB[# + 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
a[n_, m_]:= Sum[StirlingS2[n, k]*StirlingS2[k, m], {k,m,n}]; Table[a[n, m], {n, 0, 100}, {m, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2018 *)

PARI

for(n=0, 9, for(k=0, n, print1(sum(j=k, n, stirling(n, j, 2)*stirling(j, k, 2)), ", "))) \\ G. C. Greubel, Jul 10 2018

Sage

# uses[riordan_array from A256893]
riordan_array(1, exp(exp(x) - 1), 8, exp=true) # Peter Luschny, Apr 19 2015

Formula

a(n,k) = Sum_{j=k..n} S2(n,j) * S2(j,k), n>=k>=0.

E.g.f. row polynomials with argument x: exp(x*f(f(z))).

E.g.f. column k: ((exp(exp(x) - 1) - 1)^k)/k!.