A039760 -id:A039760 - cp's OEIS frontend

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

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, 268, 1, 1, 42, 581, 3255, 7371, 5747, 869, 1, 1, 56, 1092, 9296, 35686, 57568, 31060, 2768, 1, 1, 72, 1884, 22764, 134022, 373926, 441652, 166068, 8689, 1, 1, 90, 3045, 49680, 418362, 1812552, 3803290, 3342240, 879541, 26964, 1

Offset: 0

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

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

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  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, 268, 1;
  ...

Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Cf. A039760 (transposed triangle).

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]

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

More terms from Petros Hadjicostas, Jul 12 2020

A039764 D-analogs of Bell numbers.

Original entry on oeis.org

1, 1, 4, 15, 72, 403, 2546, 17867, 137528, 1149079, 10335766, 99425087, 1017259964, 11018905667, 125860969266, 1510764243699, 18999827156304, 249687992188015, 3420706820299374, 48751337014396167

Offset: 0

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

nonn

Hasan Arslan, Nazmiye Alemdar, Mariam Zaarour, and Hüseyin Altındiş, On Bell numbers of type D, arXiv:2504.16522 [math.CO], 2025. See p. 4.
Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

B-analogs of Bell numbers = A007405.

Range[0, 25]! CoefficientList[Series[(Exp[x] - x) Exp[1/2 (Exp[2 x] - 1)], {x, 0, 25}], x] (* Vincenzo Librandi, May 03 2015 *)

x='x+O('x^30); Vec(serlaplace((exp(x) - x)*exp(1/2*(exp(2*x) - 1)))) \\ Michel Marcus, May 03 2015

E.g.f.: (exp(x) - x)*exp(1/2*(exp(2*x) - 1)).

a(n) = Sum_{k=0..n} A039760(n, k).

cp's OEIS Frontend

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

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