OFFSET
0,3
COMMENTS
The Stanley reference below describes a family of binomial posets whose elements are two colored graphs with vertices labeled on [n] and with edges labeled on [k-1]. T(n,k) is the number of elements in an n-interval of such a binomial poset. - Geoffrey Critzer, Aug 21 2023
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Example 3.18.3(e), page 323.
FORMULA
E.g.f. of column k: Sum_{j>=0} exp(k^j * x) * x^j/j!.
G.f. of column k: Sum_{j>=0} x^j/(1 - k^j * x)^(j+1).
For k>=1, E(x)^2 = Sum_{n>=0} T(n,k)*x^n/B_k(n) where B_k(n) = n!*k^binomial(n,2) and E(x) = Sum_{n>=0} x^n/b_k(n). - Geoffrey Critzer, Aug 21 2023
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
2, 4, 6, 8, 10, 12, ...
2, 8, 26, 56, 98, 152, ...
2, 16, 162, 704, 2050, 4752, ...
2, 32, 1442, 15392, 84482, 318752, ...
PROG
(PARI) T(n, k) = sum(j=0, n, k^(j*(n-j))*binomial(n, j));
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 02 2022
STATUS
approved