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A156825 Generalized q-Stirling 2nd numbers (see A022166):q=4;m=3; t1(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}].

%I A156825 #19 Aug 06 2024 10:14:54
%S A156825 1,1,1,1,6,31,1,27,598,12714,1,112,10118,872744,74451015,1,453,164591,
%T A156825 56998275,19510862790,6659538174846,1,1818,2646161,3669008040,
%U A156825 5027706837390,6869479371212196,9379110782727354118,1,7279,42396780
%N A156825 Generalized q-Stirling 2nd numbers (see A022166):q=4;m=3; t1(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}].
%C A156825 Row sums are: {1, 2, 38, 13340, 75333990, 6679106200956, 9385985293477059724, 210307101689444749681505920, 75309752513141244017422009494610310, 431334730561934365895986795984802627076981452, 39523158749221869286186846414773795221687625241015791028,...}.
%H A156825 T. Kim, <a href="https://citeseerx.ist.psu.edu/pdf/eb53354f595d96ced575db8307cde35aa8f79ee4">q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients</a>, Russian Journal of Mathematical Physics, Volume 15, Number 1, March 2008, pp. 51-57, DOI:<a href="https://doi.org/10.1134/S1061920808010068">10.1134/S1061920808010068</a>.
%F A156825 t1(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}]; q=4;m=3.
%e A156825 {1},
%e A156825 {1, 1},
%e A156825 {1, 6, 31},
%e A156825 {1, 27, 598, 12714},
%e A156825 {1, 112, 10118, 872744, 74451015},
%e A156825 {1, 453, 164591, 56998275, 19510862790, 6659538174846},
%e A156825 {1, 1818, 2646161, 3669008040, 5027706837390, 6869479371212196, 9379110782727354118},
%e A156825 {1, 7279, 42396780, 235197823620, 1289443021626210, 7048517820471945006, 38501334928380019031884, 210268593304708870928675140},
%e A156825 {1, 29124, 678610560, 15059445506820, 330263030118109110, 7221644410750565452956, 157795323487774482338855704, 3447249110183738275563231529020, 75306305106228514816683123116517015},
%e A156825 {1, 116505, 10858933965, 963923954302485, 84558902081023550895, 7396063067152669466208951, 646433182194355185109143203035, 56489425142435134168297605455930355, 4936177764676230687274829745467766867270, 431329794327679618082286045004555759407867990},
%e A156825 {1, 466030, 173748069715, 61693218021437860, 21647880931024091567395, 7573871645483348274559946326, 2647903920069761660850674382798185, 925565107087525879643000261252991542480, 323513080232532159312906941144197336652189270, 113076340698070130663461880889470578649115862464740, 39523045672557657210255895794560156111877694170705309026}
%t A156825 t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
%t A156825 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156825 t1[n_, k_, q_] = (1/(q - 1)^k)*Sum[(-1)^(k - j)* Binomial[k + n, k - j]*b[j + n, j, q - 1], {j, 0, k}];
%t A156825 Table[Flatten[Table[Table[t1[n, k, m + 1], {k, 0, n}], {n, 0, 10}]], {m, 1, 15}]
%Y A156825 Cf. A022166.
%K A156825 nonn,tabl,uned
%O A156825 0,5
%A A156825 _Roger L. Bagula_, Feb 16 2009