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.

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

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..36.

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