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Search: id:A156823
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| A156823 |
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Generalized q-Stirling 2nd numbers ( see A022166):q=2;m=1; 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}]. |
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1
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| 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; listen)
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OFFSET
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0,5
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COMMENT
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Row sums are:
{1, 2, 18, 772, 131238, 82795124, 194513625552, 1734587910632112,
59780354709947486310, 8067711354683582659357588,
4300494571012469622746969756172,...}.
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REFERENCES
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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; http://www.springerlink.com/content/a76w2p508n24l60m/
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FORMULA
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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.
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EXAMPLE
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{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}
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MATHEMATICA
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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}]
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CROSSREFS
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A022166
Sequence in context: A146210 A024248 A130539 this_sequence A130650 A051432 A046737
Adjacent sequences: A156820 A156821 A156822 this_sequence A156824 A156825 A156826
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KEYWORD
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nonn,tabl,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 16 2009
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