A156823 - OEIS

A156823

Triangle T(n,k,2) read by rows (generalized q-Stirling numbers of second kind): T(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*q*Binomial[k + n, k -j] - Binomial[j + n, j, q - 1], {j, 0, k}], with q=2, where Binomial[,] is the Gaussian q-binomial coefficient as in A022166.

%I #29 Aug 06 2024 10:14:26

%S 1,1,1,1,4,13,1,11,90,670,1,26,480,7870,122861,1,57,2247,77527,

%T 2526198,80189094,1,120,9807,695368,46334382,2999255160,191467330714,

%U 1,247,41176,5924720,798773822,104443530554,13455795711072,1721026866650520,1

%N Triangle T(n,k,2) read by rows (generalized q-Stirling numbers of second kind): T(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*q*Binomial[k + n, k -j] - Binomial[j + n, j, q - 1], {j, 0, k}], with q=2, where Binomial[,] is the Gaussian q-binomial coefficient as in A022166.

%C Row sums are 1, 2, 18, 772, 131238, 82795124, 194513625552, 1734587910632112, 59780354709947486310, 8067711354683582659357588, 4300494571012469622746969756172,....

%H 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 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=2; m=1.

%e Triangle begins:

%e {1},

%e {1, 1},

%e {1, 4, 13},

%e {1, 11, 90, 670},

%e {1, 26, 480, 7870, 122861},

%e {1, 57, 2247, 77527, 2526198, 80189094},

%e {1, 120, 9807, 695368, 46334382, 2999255160, 191467330714},

%e {1, 247, 41176, 5924720, 798773822, 104443530554, 13455795711072, 1721026866650520},

%e {1, 502, 169186, 49067150, 13310897072, 3498722283914, 905629978109142, 232656671284481730, 59546788896602477613},

%e {1, 1013, 686829, 400036769, 217729686031, 114758591845755, 59547270411289947, 30661311851453644647, 15727477144989414892230, 8051953156564494657274366},

%e {1, 2036, 2769657, 3233395880, 3525493671271, 3721338617555988, 3866476676171065671, 3986066951574453826080, 4093473968605655678972070, 4195675823040150254245701976, 4296294797725523713719072795542}

%e ...

%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}]

%Y Cf. A022166.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 16 2009