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

0,5

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

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)

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!.

Example

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;

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 *)

Prog

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

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.

Sequence in context: A176056 A373224 A298213 * A054651 A292323 A059720

Adjacent sequences: A130188 A130189 A130190 * A130192 A130193 A130194

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, Jun 01 2007

Status

approved