A181783 Array described in comments to A053482, here read by increasing antidiagonals. See comments below.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 16, 21, 7, 1, 1, 1, 65, 142, 63, 11, 1, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1

Offset

0,9

Comments

We denote by a(n,k) the number in row number n >= 0 and column number k >= 0. The recurrence which defines the array is a(n,k) = n*(k-1)*a(n-1,k) + a(n,k-1). The initial values are given by a(n,0) = 1 = a(0,k) for all n >= 0 and k >= 0.

Links

Table of n, a(n) for n=0..65.

Formula

If we consider the e.g.f. Psi(k) of column number k we have: Psi(k)(z) = Psi(k-1)(z)/(1-(k-1)*z) with Psi(1)(z) = exp(z). Then Psi(k)(z) = exp(z)/Product_{j=0..k-1} (1 - j*z). We conclude that a(n,k) = n!*Sum_{m=0..n} Sum_{j=1..k-1} (-1)^(k-1-j)*j^(m+k-2)/((n-m)!*(j-1)!*(k-1-j)!). It seems after the recurrence (and its proof) in A053482 that:

A(n,k) = -Sum_{j=1..k-1} s1(k,k-j)*n*(n-1)*...*(n-k+1)*a(n-j,k) + 1 where s1(m,n) are the classical Stirling numbers of the first kind.

A(n,1) = 1 for every n.

A(1,k) = 1 + k*(k-1)/2 for every k.

A(n, k+1) = A371898(n+k, k) * n! / ((n+k)! * k!). - Werner Schulte, Apr 14 2024

Example

Array read row after row:
  1, 1,    1,      1,       1,        1,         1, ...
  1, 1,    2,      4,       7,       11,        16, ...
  1, 1,    5,     21,      63,      151,       311, ...
  1, 1,   16,    142,     709,     2521,      7186, ...
  1, 1,   65,   1201,    9709,    50045,    193765, ...
  1, 1,  326,  12336,  157971,  1158871,   6002996, ...
  1, 1, 1957, 149989, 2993467, 30806371, 210896251, ...
  ...
A(4,3) = 1201.

Maple

A181783 := proc(n, k)
    option remember;
    if n =0 or k = 0 then
        1;
    else
        n*(k-1)*procname(n-1, k)+procname(n, k-1) ;
    end if;
end proc:
seq(seq(A181783(d-k, k), k=0..d), d=0..12) ; # R. J. Mathar, Mar 02 2016

Mathematica

T[n_, k_] := T[n, k] = If[n == 0 || k == 0, 1, n (k - 1) T[n - 1, k] + T[n, k - 1]];
Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2023 *)

Crossrefs

Cf. A000522, A053482, A185106, A371898.

Sequence in context: A008326 A181196 A227578 * A121395 A275377 A219585

Adjacent sequences: A181780 A181781 A181782 * A181784 A181785 A181786

Keyword

nonn,tabl,easy

Author

Richard Choulet, Dec 23 2012

Extensions

Edited by N. J. A. Sloane, Dec 24 2012

Status

approved