A156586 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=4: m=3: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(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])].

1, 1, 1, 1, 4, 1, 1, 20, 20, 1, 1, 120, 600, 120, 1, 1, 840, 25200, 25200, 840, 1, 1, 6720, 1411200, 8467200, 1411200, 6720, 1, 1, 60480, 101606400, 4267468800, 4267468800, 101606400, 60480, 1, 1, 604800, 9144576000, 3072577536000, 21508042752000

Offset

0,5

Comments

Row sums are:

{1, 2, 6, 42, 842, 52082, 11303042, 8738271362, 27671488185602,

346773112532985602, 20244862147392528307202,...}.

The q=2 sequence is A009963.

Links

Table of n, a(n) for n=0..40.

Formula

q=4: m=3:

t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(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])].

Example

{1},
{1, 1},
{1, 4, 1},
{1, 20, 20, 1},
{1, 120, 600, 120, 1},
{1, 840, 25200, 25200, 840, 1},
{1, 6720, 1411200, 8467200, 1411200, 6720, 1},
{1, 60480, 101606400, 4267468800, 4267468800, 101606400, 60480, 1},
{1, 604800, 9144576000, 3072577536000, 21508042752000, 3072577536000, 9144576000, 604800, 1},
{1, 6652800, 1005903360000, 3041851760640000, 170343698595840000, 170343698595840000, 3041851760640000, 1005903360000, 6652800, 1},
{1, 79833600, 132779243520000, 4015244324044800000, 2023683139318579200000, 16189465114548633600000, 2023683139318579200000, 4015244324044800000, 132779243520000, 79833600, 1}

Mathematica

Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(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])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

Crossrefs

A009963

Sequence in context: A254442 A340476 A176422 * A181544 A154283 A185946

Adjacent sequences: A156583 A156584 A156585 * A156587 A156588 A156589

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 10 2009

Status

approved