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A039760
Triangle of D-analogs of Stirling numbers of the 2nd kind.
3
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
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
KEYWORD
nonn,tabl
AUTHOR
Ruedi Suter (suter(AT)math.ethz.ch)
STATUS
approved