A156824 - OEIS

%I #16 Mar 09 2020 09:14:24

%S 1,1,1,1,5,21,1,18,255,3400,1,58,2575,106400,4300541,1,179,24234,

%T 3038714,371984935,45182779173,1,543,221886,83805218,30877084287,

%U 11284441459641,4113010719221412,1,1636,2010034,2280772380,2523761295627,2769755537579952,3031455813294108948,3314879002466198503080

%N Generalized q-Stirling 2nd numbers (see A022166):q=3;m=2; 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 Row sums are: {1, 2, 27, 3674, 4409575, 45557827236, 4124326121792988, 3317913230561074271658, 23891408190421363405102296351, 1544865931069396100350109616919010834, 898255701914264060744770399113246348926078875,...}.

%H T. Kim, <a href="https://doi.org/10.1134/S1061920808010068">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.

%F 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=3;m=2.

%e {1},

%e {1, 1},

%e {1, 5, 21},

%e {1, 18, 255, 3400},

%e {1, 58, 2575, 106400, 4300541},

%e {1, 179, 24234, 3038714, 371984935, 45182779173},

%e {1, 543, 221886, 83805218, 30877084287, 11284441459641, 4113010719221412},

%e {1, 1636, 2010034, 2280772380, 2523761295627, 2769755537579952, 3031455813294108948, 3314879002466198503080},

%e {1, 4916, 18134514, 61761978300, 205103050119627, 675507759929956512, 2218696908383551468308, 7280640738500515014553320, 23884125330310241581776080853},

%e {1, 14757, 163358151, 1669369542291, 16633368715805358, 164364489292170484590, 1619729636032633290318498, 15947039988935644725038892138, 156958704656445989980689513610911, 1544708956416079771407327238656984139},

%e {1, 44281, 1470710395, 45090623244271, 1347888929379662362, 39959437240297322060278, 1181382154718570769797966170, 34895073775900019052240192095218, 1030399116864328608488320120932827143, 30423048235258853916780570577659445554071, 898225277835594788771326994531551239761914685}

%t t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

%t b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

%t 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 Table[Flatten[Table[Table[t1[n, k, m + 1], {k, 0, n}], {n, 0, 10}]], {m, 1, 15}]

%t (* Second program: *)

%t (* S stands for qStirling2 *)

%t S[n_, k_, q_] /; 1 <= k <= n := S[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}]* S[n-1, k, q];

%t S[n_, 0, _] := KroneckerDelta[n, 0];

%t S[0, k_, _] := KroneckerDelta[0, k];

%t S[_, _, _] = 0;

%t Table[S[n+k, n, 3], {n, 0, 7}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 09 2020 *)

%Y Cf. A022166.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 16 2009