A341102 T(n, k) = [n, k] - {n, k}, where [n, k] are the (unsigned) Stirling cycle numbers and {n, k} the Stirling set numbers. Table T(n, k) read by rows, for n >= 3 and 1 <= k <= n-2.

1, 5, 4, 23, 35, 10, 119, 243, 135, 20, 719, 1701, 1323, 385, 35, 5039, 12941, 12166, 5068, 910, 56, 40319, 109329, 115099, 59514, 15498, 1890, 84, 362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120, 3628799, 10627617, 12725075, 8263750, 3170200, 722568, 93786, 6270, 165

Offset

3,2

Links

Table of n, a(n) for n=3..47.

Peter Luschny, The difference of the Stirling cycle numbers and the Stirling set numbers, Mathematics Stack Exchange, Feb. 2021.

Formula

T(n, k) = Sum_{j=0..k} (binomial(n+j-1, 2*k) - binomial(n+k-j, 2*k))*A340556(k, j).

E.g.f.: (1 - z)^(-x) - exp(x*(exp(z) - 1)) (unrestricted rows and n >= 0).

Example

Triangle starts:
[ 3] [1]
[ 4] [5,      4]
[ 5] [23,     35,      10]
[ 6] [119,    243,     135,     20]
[ 7] [719,    1701,    1323,    385,    35]
[ 8] [5039,   12941,   12166,   5068,   910,    56]
[ 9] [40319,  109329,  115099,  59514,  15498,  1890,  84]
[10] [362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120]

Maple

# Giving full rows for n >= 0:
gf := (1 - z)^(-x) - exp(x*(exp(z) - 1));
ser := series(gf, z, 20): coeffz := n -> coeff(ser, z, n):
A341102row := n -> seq(n!*coeff(coeffz(n), x, k), k=0..n):
for n from 0 to 9 do A341102row(n) od;

Prog

(SageMath)
for n in (3..11):
    print([stirling_number1(n, k) - stirling_number2(n, k) for k in (1..n-2)])
(PARI) T(n, k) = abs(stirling(n, k, 1)) - stirling(n, k, 2); \\ Michel Marcus, Feb 24 2021

Crossrefs

Cf. A132393, A048993, A048742, A033312, A000292, A340556.

Sequence in context: A240987 A338509 A204930 * A167636 A180137 A288207

Adjacent sequences: A341099 A341100 A341101 * A341103 A341104 A341105

Keyword

nonn,tabl

Author

Peter Luschny, Feb 24 2021

Status

approved