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 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. 0
 1, 1, 1, 1, 4, 13, 1, 11, 90, 670, 1, 26, 480, 7870, 122861, 1, 57, 2247, 77527, 2526198, 80189094, 1, 120, 9807, 695368, 46334382, 2999255160, 191467330714, 1, 247, 41176, 5924720, 798773822, 104443530554, 13455795711072, 1721026866650520, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are 1, 2, 18, 772, 131238, 82795124, 194513625552, 1734587910632112, 59780354709947486310, 8067711354683582659357588, 4300494571012469622746969756172,.... 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, DOI:10.1134/S1061920808010068. 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=2; m=1. EXAMPLE Triangle begins: {1}, {1, 1}, {1, 4, 13}, {1, 11, 90, 670}, {1, 26, 480, 7870, 122861}, {1, 57, 2247, 77527, 2526198, 80189094}, {1, 120, 9807, 695368, 46334382, 2999255160, 191467330714}, {1, 247, 41176, 5924720, 798773822, 104443530554, 13455795711072, 1721026866650520}, {1, 502, 169186, 49067150, 13310897072, 3498722283914, 905629978109142, 232656671284481730, 59546788896602477613}, {1, 1013, 686829, 400036769, 217729686031, 114758591845755, 59547270411289947, 30661311851453644647, 15727477144989414892230, 8051953156564494657274366}, {1, 2036, 2769657, 3233395880, 3525493671271, 3721338617555988, 3866476676171065671, 3986066951574453826080, 4093473968605655678972070, 4195675823040150254245701976, 4296294797725523713719072795542} ... 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}] CROSSREFS Cf. A022166. Sequence in context: A146210 A024248 A130539 * A212256 A265327 A130650 Adjacent sequences:  A156820 A156821 A156822 * A156824 A156825 A156826 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 16 2009 STATUS approved

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Last modified January 27 19:46 EST 2021. Contains 340479 sequences. (Running on oeis4.)