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A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.

%I A333143 #23 Mar 11 2020 08:25:53
%S A333143 1,1,1,1,5,1,1,21,18,1,1,85,255,58,1,1,341,3400,2575,179,1,1,1365,
%T A333143 44541,106400,24234,543,1,1,5461,580398,4300541,3038714,221886,1636,1,
%U A333143 1,21845,7550635,172602038,371984935,83805218,2010034,4916,1
%N A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.
%F A333143 qStirling2(n, k, q) = qStirling2(n-1, k-1, q) + qBrackets(k+1, q)*qStirling2(n-1, k, q) with boundary values 0^k if n = 0 and n^0 if k = 0.
%F A333143 Note that also a second definition is used in the literature which has an additional factor q^k attached to the first term in the equation above. The two versions differ by a factor of q^binomial(k,2).
%e A333143 [0] 1
%e A333143 [1] 1, 1
%e A333143 [2] 1, 5,     1
%e A333143 [3] 1, 21,    18,      1
%e A333143 [4] 1, 85,    255,     58,        1
%e A333143 [5] 1, 341,   3400,    2575,      179,       1
%e A333143 [6] 1, 1365,  44541,   106400,    24234,     543,      1
%e A333143 [7] 1, 5461,  580398,  4300541,   3038714,   221886,   1636,    1
%e A333143 [8] 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1
%p A333143 qStirling2 := proc(n, k, q) option remember; with(QDifferenceEquations):
%p A333143 if n = 0 then return 0^k fi; if k = 0 then return n^0 fi;
%p A333143 qStirling2(n-1, k-1, p) + QBrackets(k+1, p)*qStirling2(n-1, k, p);
%p A333143 subs(p = q, expand(%)) end:
%p A333143 seq(seq(qStirling2(n, k, 3), k=0..n), n=0..9);
%t A333143 qStirling2[n_, k_, q_] /; 1 <= k <= n := (* q^(k-1) *) qStirling2[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}] qStirling2[n - 1, k, q];
%t A333143 qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
%t A333143 qStirling2[0, k_, _] := KroneckerDelta[0, k];
%t A333143 qStirling2[_, _, _] = 0;
%t A333143 Table[qStirling2[n + 1, k + 1, 3], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 11 2020 *)
%Y A333143 T(n, 1) = A002450(n), T(n, n-1) = A000340(n).
%Y A333143 Cf. A139382 (q=2), A333142.
%K A333143 nonn,tabl
%O A333143 0,5
%A A333143 _Peter Luschny_, Mar 09 2020