A059340 Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.

1, 2, 1, 5, 5, 1, 15, 23, 10, 1, 52, 109, 76, 19, 1, 203, 544, 531, 224, 36, 1, 877, 2876, 3641, 2204, 631, 69, 1, 4140, 16113, 25208, 20089, 8471, 1749, 134, 1, 21147, 95495, 178564, 177631, 100171, 31331, 4838, 263, 1

Offset: 1

Author

Vladeta Jovovic, Jan 27 2001

Keywords

• easy
• nonn
• tabl

Comments

Essentially triangle given by [1,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [0,1,0,2,0,3,0,4,0,5,0,6,...] = [1;1,0;2,1,0;5,5,1,0;15,23,10,1,0;...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 20 2006

Examples

			Triangle starts:
   1;
   2,   1;
   5,   5,   1;
  15,  23,  10,   1;
  52, 109,  76,  19,   1;
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums = A004211, T(n,0) = A000110.

Programs

Mathematica

Table[Sum[StirlingS2[n, j]*Binomial[n - j, k], {j, 0, n}], {n, 1,
  5}, {k, 0, n - 1}] (* G. C. Greubel, Jan 07 2017 *)

Sage

T = lambda n,k: sum(stirling_number2(n,j)*binomial(n-j,k) for j in (0..n))
# Also "for n in (0..11): print([T(n,k) for k in (0..n)])" makes sense.
for n in (1..11): print([T(n,k) for k in (0..n-1)]) # Peter Luschny, Aug 06 2015

Formula

T(n,k) = Sum_{i=0..n} stirling2(n, n-i)*binomial(i, k).

T(n,k) = Sum_{i=0..n} stirling2(n, i)*binomial(n-i, k). - Peter Luschny, Aug 06 2015