A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)*Bell(k)*Bell(n-k).

1, 1, 1, 2, 2, 2, 5, 6, 6, 5, 15, 20, 24, 20, 15, 52, 75, 100, 100, 75, 52, 203, 312, 450, 500, 450, 312, 203, 877, 1421, 2184, 2625, 2625, 2184, 1421, 877, 4140, 7016, 11368, 14560, 15750, 14560, 11368, 7016, 4140, 21147, 37260, 63144, 85260, 98280, 98280, 85260, 63144, 37260, 21147

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Examples

			   1;
   1,  1;
   2,  2,   2;
   5,  6,   6,   5;
  15, 20,  24,  20, 15;
  52, 75, 100, 100, 75, 52;
  ...
		

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 80.

Crossrefs

Cf. A000110, row sums give A001861.

Columns include A000110 and A052889.

Cf. A000807.

Programs

Maple

A033306 := proc(n,k)
    if k < 0 or k > n then
        0;
    else
        binomial(n,k)*combinat[bell](k)*combinat[bell](n-k) ;
    end if;
end proc: # R. J. Mathar, Mar 21 2013
# second Maple program:
b:= proc(n) option remember; expand(`if`(n>0, add(
     (x^j+1)*b(n-j)*binomial(n-1, j-1), j=1..n), 1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 30 2019

Mathematica

t[n_, k_] := Binomial[n, k] * BellB[k] * BellB[n-k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Formula

E.g.f.: exp(exp(x*y)+exp(x)-2).

Sum_{k=0..2n} (-1)^k * T(2n,k) = A000807(n). - Alois P. Heinz, Feb 13 2024

Extensions

Edited by Vladeta Jovovic, Sep 17 2003