A039760 Triangle of D-analogs of Stirling numbers of the 2nd kind.

1, 0, 1, 1, 2, 1, 1, 7, 6, 1, 1, 24, 34, 12, 1, 1, 81, 190, 110, 20, 1, 1, 268, 1051, 920, 275, 30, 1, 1, 869, 5747, 7371, 3255, 581, 42, 1, 1, 2768, 31060, 57568, 35686, 9296, 1092, 56, 1, 1, 8689, 166068, 441652, 373926, 134022, 22764, 1884, 72, 1

Offset: 0

Author

Ruedi Suter (suter(AT)math.ethz.ch)

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k=0..n) begins:
  1;
  0,   1;
  1,   2,    1;
  1,   7,    6,   1;
  1,  24,   34,  12,   1;
  1,  81,  190, 110,  20,  1;
  1, 268, 1051, 920, 275, 30, 1;
  ...
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• Eli Bagno, Riccardo Biagioli, and David Garber, Some identities involving second kind Stirling numbers of types B and D, arXiv:1901.07830 [math.CO], 2019.
• Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Crossrefs

Cf. A039761 (transposed triangle).

Programs

Mathematica

With[{m = 10}, CoefficientList[CoefficientList[Series[(Exp[x]-x)* Exp[y/2*(Exp[2*x]-1)], {y, 0, m}, {x, 0, m}], x], y]*(Range[0, m]!)] (* G. C. Greubel, Mar 07 2019 *)

PARI

T(n, k)=if(k<0||k>n, 0, n!*polcoeff(polcoeff((exp(x)-x)*exp(y/2*(exp(2*x)-1)), n), k));
tabl(nn) = {x = 'x + O('x^nn); for (n=0, nn, for (m=0, n, print1(T(n, m), ", ");); print(););} \\ Michel Marcus, May 03 2015

Formula

Bivariate e.g.f.-o.g.f.: (exp(x) - x)*exp(y/2*(exp(2*x) - 1)). [See Theorem 4 in Suter (2000).]

T(n,k) = Sum_{j=k..n} 2^(j-k)*binomial(n,j)*Stirling2(j,k) - 2^(n-1-k)*n*Stirling2(n-1,k). [See Proposition 3 in Suter (2000).] - Petros Hadjicostas, Jul 11 2020