A363007 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - f^k(x)), where f(x) = exp(x) - 1.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 24, 1, 1, 5, 23, 75, 120, 1, 1, 6, 36, 175, 541, 720, 1, 1, 7, 52, 342, 1662, 4683, 5040, 1, 1, 8, 71, 594, 4048, 18937, 47293, 40320, 1, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 362880, 1, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 3628800

Offset

0,6

Links

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

Formula

T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.

T(n,k) = A153278(k,n) for n >= 1 and k >= 1.

Example

Square array begins:
    1,   1,    1,    1,    1,     1, ...
    1,   1,    1,    1,    1,     1, ...
    2,   3,    4,    5,    6,     7, ...
    6,  13,   23,   36,   52,    71, ...
   24,  75,  175,  342,  594,   949, ...
  120, 541, 1662, 4048, 8444, 15775, ...

Prog

(PARI) T(n, k) = if(k==0, n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));

Crossrefs

Columns k=0..5 give A000142, A000670, A083355, A099391, A363008, A363009.

Main diagonal gives A363010.

Cf. A153278, A351420, A351429.

Sequence in context: A128325 A307883 A111528 * A144303 A370072 A287024

Adjacent sequences: A363004 A363005 A363006 * A363008 A363009 A363010

Keyword

nonn,tabl

Author

Seiichi Manyama, May 12 2023

Status

approved