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

%I A333142 #11 Mar 09 2020 13:10:37
%S A333142 1,1,1,1,2,1,1,7,5,1,1,50,42,12,1,1,751,680,222,27,1,1,23282,21831,
%T A333142 7562,1059,58,1,1,1466767,1398635,498237,74279,4713,121,1,1,186279410,
%U A333142 179093412,64674734,9931670,672830,20080,248,1
%N A333142 Triangle read by rows: T(n, k) = qStirling1(n, k, q) for q = 2, with 0 <= k <= n.
%F A333142 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.
%F A333142 Note that also a second definition is used in the literature. The two versions differ by a factor of q^(n-k).
%e A333142 Triangle starts:
%e A333142 [0] 1
%e A333142 [1] 1, 1
%e A333142 [2] 1, 2,         1
%e A333142 [3] 1, 7,         5,         1
%e A333142 [4] 1, 50,        42,        12,       1
%e A333142 [5] 1, 751,       680,       222,      27,      1
%e A333142 [6] 1, 23282,     21831,     7562,     1059,    58,     1
%e A333142 [7] 1, 1466767,   1398635,   498237,   74279,   4713,   121,   1
%e A333142 [8] 1, 186279410, 179093412, 64674734, 9931670, 672830, 20080, 248, 1
%p A333142 qStirling1 := proc(n, k, q) option remember; with(QDifferenceEquations):
%p A333142 if n = 0 then return 0^k fi; if k = 0 then return n^0 fi;
%p A333142 qStirling1(n-1, k-1, p) + QBrackets(n-1, p)*qStirling1(n-1, k, p);
%p A333142 subs(p = q, expand(%)) end:
%p A333142 seq(seq(qStirling1(n, k, 2), k=0..n), n=0..9);
%Y A333142 T(n,n-1) = A000325(n).
%Y A333142 Cf. A333143.
%K A333142 nonn,tabl
%O A333142 0,5
%A A333142 _Peter Luschny_, Mar 09 2020