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

0,5

Links

Table of n, a(n) for n=0..44.

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.

Formula

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

Example

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

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 *)

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).

Sequence in context: A156278 A166961 A202988 * A022172 A173005 A015123

Adjacent sequences: A098433 A098434 A098435 * A098437 A098438 A098439

Keyword

tabl,nonn

Author

Ralf Stephan, Sep 08 2004

Status

approved