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

0,5

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.

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

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

Cf. A039761 (transposed triangle).

Sequence in context: A333142 A196832 A005450 * A156882 A266309 A122021

Adjacent sequences: A039757 A039758 A039759 * A039761 A039762 A039763

Keyword

nonn,tabl

Author

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

Status

approved