A039761 Triangle of D-analogs of Stirling numbers of 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

0,5

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

Links

Table of n, a(n) for n=0..65.

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*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

Example

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;
  ...

Crossrefs

Cf. A039760 (transposed triangle).

Sequence in context: A173887 A288025 A137376 * A196073 A144089 A172107

Adjacent sequences: A039758 A039759 A039760 * A039762 A039763 A039764

Keyword

nonn,tabl

Author

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

Extensions

More terms from Petros Hadjicostas, Jul 12 2020

Status

approved