A061692 Triangle of generalized Stirling numbers.

1, 1, 4, 1, 27, 36, 1, 172, 864, 576, 1, 1125, 17500, 36000, 14400, 1, 7591, 351000, 1746000, 1944000, 518400, 1, 52479, 7197169, 80262000, 191394000, 133358400, 25401600, 1, 369580, 151633440, 3691514176, 17188416000, 23866214400, 11379916800, 1625702400

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

Diagonals give A001044, A061695, A061693, A061694. Cf. A061691.

Row sums give A061684.

Sequence in context: A225213 A137906 A139051 * A096206 A336913 A134150

Adjacent sequences: A061689 A061690 A061691 * A061693 A061694 A061695

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 19 2001

Extensions

More terms from Vladeta Jovovic, Apr 23 2003

Status

approved