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
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
KEYWORD
nonn,tabl
AUTHOR
Ruedi Suter (suter(AT)math.ethz.ch)
EXTENSIONS
More terms from Petros Hadjicostas, Jul 12 2020
STATUS
approved