A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

1, 3, 1, 14, 9, 1, 90, 83, 18, 1, 744, 870, 275, 30, 1, 7560, 10474, 4275, 685, 45, 1, 91440, 143892, 70924, 14805, 1435, 63, 1, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578

Offset: 1

Author

Vladeta Jovovic, Jan 30 2003

Keywords

• nonn
• tabl

Comments

Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.

Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Essentially the same as A131222. - Peter Bala, Feb 12 2022

Examples

			Triangle begins
     1;
     3,     1;
    14,     9,    1;
    90,    83,   18,   1;
   744,   870,  275,  30,  1;
  7560, 10474, 4275, 685, 45, 1;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1225 (rows n = 1..50, flattened).
• William Keith, Rishi Nath, and James Sellers, On simultaneous (s, s+t, s+2t, ...)-core partitions, arXiv:2508.00074 [math.CO], 2025. See p. 3.

Crossrefs

Cf. A002866 (row sums), A029767 (first column), A131222.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> n!*(2^(n+1)-1), 9); # Peter Luschny, Jan 26 2016

Mathematica

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Formula

T(n, k) = Sum_{i=k..n} |A008297(n, i)| * |A008275(i, k)|.

E.g.f.: ((1-x)/(1-2*x))^y. - Vladeta Jovovic, Nov 22 2003