A333142 Triangle read by rows: T(n, k) = qStirling1(n, k, q) for q = 2, with 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 7, 5, 1, 1, 50, 42, 12, 1, 1, 751, 680, 222, 27, 1, 1, 23282, 21831, 7562, 1059, 58, 1, 1, 1466767, 1398635, 498237, 74279, 4713, 121, 1, 1, 186279410, 179093412, 64674734, 9931670, 672830, 20080, 248, 1

Offset: 0

Author

Peter Luschny, Mar 09 2020

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1
[1] 1, 1
[2] 1, 2,         1
[3] 1, 7,         5,         1
[4] 1, 50,        42,        12,       1
[5] 1, 751,       680,       222,      27,      1
[6] 1, 23282,     21831,     7562,     1059,    58,     1
[7] 1, 1466767,   1398635,   498237,   74279,   4713,   121,   1
[8] 1, 186279410, 179093412, 64674734, 9931670, 672830, 20080, 248, 1
		

Crossrefs

T(n,n-1) = A000325(n).

Cf. A333143.

Programs

Maple

qStirling1 := proc(n, k, q) option remember; with(QDifferenceEquations):
if n = 0 then return 0^k fi; if k = 0 then return n^0 fi;
qStirling1(n-1, k-1, p) + QBrackets(n-1, p)*qStirling1(n-1, k, p);
subs(p = q, expand(%)) end:
seq(seq(qStirling1(n, k, 2), k=0..n), n=0..9);

Formula

qStirling1(n, k, q) = qStirling1(n-1, k-1, q) + qBrackets(n-1, q)*qStirling1(n-1, k, q) with boundary values 0^k if n = 0 and n^0 if k = 0.

Note that also a second definition is used in the literature. The two versions differ by a factor of q^(n-k).