The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A156825 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}]. 0
 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; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 38, 13340, 75333990, 6679106200956, 9385985293477059724, 210307101689444749681505920, 75309752513141244017422009494610310, 431334730561934365895986795984802627076981452, 39523158749221869286186846414773795221687625241015791028,...}. 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=4;m=3. EXAMPLE {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} 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: A119411 A036285 A101340 * A043058 A155097 A306331 Adjacent sequences:  A156822 A156823 A156824 * A156826 A156827 A156828 KEYWORD nonn,tabl,uned,changed AUTHOR Roger L. Bagula, Feb 16 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 30 11:06 EST 2020. Contains 338799 sequences. (Running on oeis4.)