OFFSET
1,3
LINKS
Alois P. Heinz, Rows n = 1..100, flattened
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
FORMULA
T(n, k) = 1/k!*Sum multinomial(n, n_1, n_2, ..n_k)^3, where the sum extends over all compositions (n_1, n_2, .., n_k) of n into exactly k nonnegative parts. - Vladeta Jovovic, Apr 23 2003
The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*( sum {k = 0..n-1} binomial(n,k)^2*binomial(n-1,k)*R(k,x) ) with R(0,x) = 1. Also R(n,x + y) = sum {k = 0..n} binomial(n,k)^3*R(k,x)*R(n-k,y). - Peter Bala, Sep 17 2013
EXAMPLE
1; 1,4; 1,27,36; 1,172,864,576; ...
MAPLE
b:= proc(n) option remember; expand(
`if`(n=0, 1, add(x*b(n-i)/i!^3, i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!^3):
seq(T(n), n=1..10); # Alois P. Heinz, Sep 10 2019
MATHEMATICA
R[0, _] = 1; R[n_, x_] := R[n, x] = x*Sum[Binomial[n, k]^2*Binomial[n-1, k]*R[k, x], {k, 0, n-1}]; Table[CoefficientList[R[n, x], x] // Rest, {n, 1, 8}] // Flatten (* Jean-François Alcover, Sep 01 2015, after Peter Bala *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 19 2001
EXTENSIONS
More terms from Vladeta Jovovic, Apr 23 2003
STATUS
approved