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

%I A039760 #32 Jul 13 2020 02:18:58
%S A039760 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,
%T A039760 275,30,1,1,869,5747,7371,3255,581,42,1,1,2768,31060,57568,35686,9296,
%U A039760 1092,56,1,1,8689,166068,441652,373926,134022,22764,1884,72,1
%N A039760 Triangle of D-analogs of Stirling numbers of the 2nd kind.
%H A039760 G. C. Greubel, <a href="/A039760/b039760.txt">Rows n=0..100 of triangle, flattened</a>
%H A039760 Eli Bagno, Riccardo Biagioli, and David Garber, <a href="https://arxiv.org/abs/1901.07830">Some identities involving second kind Stirling numbers of types B and D</a>, arXiv:1901.07830 [math.CO], 2019.
%H A039760 Ruedi Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
%F A039760 Bivariate e.g.f.-o.g.f.: (exp(x) - x)*exp(y/2*(exp(2*x) - 1)). [See Theorem 4 in Suter (2000).]
%F A039760 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
%e A039760 Triangle T(n,k) (with rows n >= 0 and columns k=0..n) begins:
%e A039760   1;
%e A039760   0,   1;
%e A039760   1,   2,    1;
%e A039760   1,   7,    6,   1;
%e A039760   1,  24,   34,  12,   1;
%e A039760   1,  81,  190, 110,  20,  1;
%e A039760   1, 268, 1051, 920, 275, 30, 1;
%e A039760   ...
%t A039760 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 *)
%o A039760 (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));
%o A039760 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
%Y A039760 Cf. A039761 (transposed triangle).
%K A039760 nonn,tabl
%O A039760 0,5
%A A039760 Ruedi Suter (suter(AT)math.ethz.ch)