A156824 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}].

1, 1, 1, 1, 5, 21, 1, 18, 255, 3400, 1, 58, 2575, 106400, 4300541, 1, 179, 24234, 3038714, 371984935, 45182779173, 1, 543, 221886, 83805218, 30877084287, 11284441459641, 4113010719221412, 1, 1636, 2010034, 2280772380, 2523761295627, 2769755537579952, 3031455813294108948, 3314879002466198503080

Offset: 0

Author

Roger L. Bagula, Feb 16 2009

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 27, 3674, 4409575, 45557827236, 4124326121792988, 3317913230561074271658, 23891408190421363405102296351, 1544865931069396100350109616919010834, 898255701914264060744770399113246348926078875,...}.

Examples

			{1},
{1, 1},
{1, 5, 21},
{1, 18, 255, 3400},
{1, 58, 2575, 106400, 4300541},
{1, 179, 24234, 3038714, 371984935, 45182779173},
{1, 543, 221886, 83805218, 30877084287, 11284441459641, 4113010719221412},
{1, 1636, 2010034, 2280772380, 2523761295627, 2769755537579952, 3031455813294108948, 3314879002466198503080},
{1, 4916, 18134514, 61761978300, 205103050119627, 675507759929956512, 2218696908383551468308, 7280640738500515014553320, 23884125330310241581776080853},
{1, 14757, 163358151, 1669369542291, 16633368715805358, 164364489292170484590, 1619729636032633290318498, 15947039988935644725038892138, 156958704656445989980689513610911, 1544708956416079771407327238656984139},
{1, 44281, 1470710395, 45090623244271, 1347888929379662362, 39959437240297322060278, 1181382154718570769797966170, 34895073775900019052240192095218, 1030399116864328608488320120932827143, 30423048235258853916780570577659445554071, 898225277835594788771326994531551239761914685}
		

Links

• T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russian Journal of Mathematical Physics, Volume 15, Number 1, March 2008, pp. 51-57.

Crossrefs

Cf. A022166.

Programs

Mathematica

t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
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}];
Table[Flatten[Table[Table[t1[n, k, m + 1], {k, 0, n}], {n, 0, 10}]], {m, 1, 15}]
(* Second program: *)
(* S stands for qStirling2 *)
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];
S[n_, 0, _] := KroneckerDelta[n, 0];
S[0, k_, _] := KroneckerDelta[0, k];
S[, , _] = 0;
Table[S[n+k, n, 3], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 09 2020 *)

Formula

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.