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A039761 Triangle of D-analogs of Stirling numbers of the 2nd kind.

%I A039761 #38 Jan 03 2025 23:29:44
%S A039761 1,1,0,1,2,1,1,6,7,1,1,12,34,24,1,1,20,110,190,81,1,1,30,275,920,1051,
%T A039761 268,1,1,42,581,3255,7371,5747,869,1,1,56,1092,9296,35686,57568,31060,
%U A039761 2768,1,1,72,1884,22764,134022,373926,441652,166068,8689,1,1,90,3045,49680,418362,1812552,3803290,3342240,879541,26964,1
%N A039761 Triangle of D-analogs of Stirling numbers of the 2nd kind.
%C A039761 Since T(n,k) = A039760(n,n-k), we have Sum_{n,k >= 0} T(n,k)*(x^n/n!)*y^k = Sum_{n,k >= 0} A039760(n,n-k)*((x*y)^n/n!)*(1/y)^(n-k) = Sum_{n,m >= 0} A039760(n,m)*((x*y)^n/n!)*(1/y)^m. Thus, to get the bivariate e.g.f.-o.g.f. of T(n,k), we perform the following transformation in the bivariate e.g.f.-o.g.f. of A039760: (x,y) -> (x*y, 1/y). - _Petros Hadjicostas_, Jul 11 2020
%H A039761 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 A039761 Bivariate e.g.f.-o.g.f.: (exp(x*y) - x*y) * exp(1/(2*y)*(exp(2*x*y) - 1)). [Apply (x, y) -> (x*y, 1/y) to (exp(x) - x)*exp(y/2*(exp(2*x) - 1)). - _Petros Hadjicostas_, Jul 11 2020]
%F A039761 T(n,k) = (Sum_{j=n-k..n} 2^(j+k-n)*binomial(n,j)*Stirling2(j, n-k)) - 2^(k-1)*n*Stirling2(n-1, n-k). [Use Proposition 3 in Suter (2000) with k -> n-k.] - _Petros Hadjicostas_, Jul 11 2020
%e A039761 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A039761   1;
%e A039761   1,  0;
%e A039761   1,  2,   1;
%e A039761   1,  6,   7,   1;
%e A039761   1, 12,  34,  24,    1;
%e A039761   1, 20, 110, 190,   81,   1;
%e A039761   1, 30, 275, 920, 1051, 268, 1;
%e A039761   ...
%Y A039761 Cf. A039760 (transposed triangle).
%K A039761 nonn,tabl
%O A039761 0,5
%A A039761 Ruedi Suter (suter(AT)math.ethz.ch)
%E A039761 More terms from _Petros Hadjicostas_, Jul 12 2020