A098436 Triangle of 3rd central factorial numbers T(n,k).

1, 1, 1, 1, 9, 1, 1, 73, 36, 1, 1, 585, 1045, 100, 1, 1, 4681, 28800, 7445, 225, 1, 1, 37449, 782281, 505280, 35570, 441, 1, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1, 1, 2396745, 571593565, 2140851900, 652061451, 33209946, 399738, 1296, 1

Offset: 0

Author

Ralf Stephan, Sep 08 2004

Keywords

• tabl
• nonn

Examples

			  1;
  1,   1;
  1,   9,    1;
  1,  73,   36,   1;
  1, 585, 1045, 100, 1;
  ...
		

Links

• M. Domaratzki, A Generalization of the Genocchi Numbers with Applications to Enumeration of Finite Automata, Technical Report No. 2001-449, September 2001, 11 pages.
• John Riordan, Letter, Apr 28 1976.

Crossrefs

First column is A023001, first diagonal is A000537.

Row sums are in A098437.

Replace in recurrence (k+1)^3 with k: A008277; with k^2: A008957 (note offsets).

Programs

Maple

A098436 := proc(n,k)
    option remember;
    if k=0 or k = n then
        1;
    else
        (k+1)^3*procname(n-1,k)+procname(n-1,k-1) ;
    end if;
end proc:
seq(seq( A098436(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jan 13 2025

Mathematica

T[n_, n_] = 1;
T[n_ /; n>=0, k_] /; 0<=k<=n := T[n, k] = (k+1)^3 T[n-1, k]+T[n-1, k-1];
T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 08 2022 *)

Formula

Recurrence: T(n, k) = (k+1)^3*T(n-1, k) + T(n-1, k-1), T(0, 0)=1.