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

Author

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

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A039760 (transposed triangle).

Formula

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

Extensions

More terms from Petros Hadjicostas, Jul 12 2020