A355395 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(j*(n-j)) * binomial(n,j).

1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 6, 8, 2, 1, 2, 8, 26, 16, 2, 1, 2, 10, 56, 162, 32, 2, 1, 2, 12, 98, 704, 1442, 64, 2, 1, 2, 14, 152, 2050, 15392, 18306, 128, 2, 1, 2, 16, 218, 4752, 84482, 593408, 330626, 256, 2, 1, 2, 18, 296, 9506, 318752, 7221250, 39691136, 8488962, 512, 2

Offset: 0

Author

Seiichi Manyama, Jul 02 2022

Keywords

• nonn
• tabl

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

Examples

			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, ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Example 3.18.3(e), page 323.

Crossrefs

Columns k=0..4 give A040000, A000079, A047863, A135079, A355440.

Main diagonal gives A320287.

Cf. A009999.

Programs

PARI

T(n, k) = sum(j=0, n, k^(j*(n-j))*binomial(n, j));

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