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Search: id:A156825
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| A156825 |
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Generalized q-Stirling 2nd numbers ( see A022166):q=4;m=3; 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, 6, 31, 1, 27, 598, 12714, 1, 112, 10118, 872744, 74451015, 1, 453, 164591, 56998275, 19510862790, 6659538174846, 1, 1818, 2646161, 3669008040, 5027706837390, 6869479371212196, 9379110782727354118, 1, 7279, 42396780
(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, 38, 13340, 75333990, 6679106200956, 9385985293477059724,
210307101689444749681505920, 75309752513141244017422009494610310,
431334730561934365895986795984802627076981452,
39523158749221869286186846414773795221687625241015791028,...}.
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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=4;m=3.
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EXAMPLE
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{1},
{1, 1},
{1, 6, 31},
{1, 27, 598, 12714},
{1, 112, 10118, 872744, 74451015},
{1, 453, 164591, 56998275, 19510862790, 6659538174846},
{1, 1818, 2646161, 3669008040, 5027706837390, 6869479371212196, 9379110782727354118},
{1, 7279, 42396780, 235197823620, 1289443021626210, 7048517820471945006, 38501334928380019031884, 210268593304708870928675140},
{1, 29124, 678610560, 15059445506820, 330263030118109110, 7221644410750565452956, 157795323487774482338855704, 3447249110183738275563231529020, 75306305106228514816683123116517015},
{1, 116505, 10858933965, 963923954302485, 84558902081023550895, 7396063067152669466208951, 646433182194355185109143203035, 56489425142435134168297605455930355, 4936177764676230687274829745467766867270, 431329794327679618082286045004555759407867990},
{1, 466030, 173748069715, 61693218021437860, 21647880931024091567395, 7573871645483348274559946326, 2647903920069761660850674382798185, 925565107087525879643000261252991542480, 323513080232532159312906941144197336652189270, 113076340698070130663461880889470578649115862464740, 39523045672557657210255895794560156111877694170705309026}
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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: A119411 A036285 A101340 this_sequence A043058 A155097 A025524
Adjacent sequences: A156822 A156823 A156824 this_sequence A156826 A156827 A156828
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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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