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A156581 A q combination generalization of Paul Hanna's general sequence in A118180, A118185, A118190, etc.: here q=16:m=15; t(n,m)=If[m == 0, n!, Product[Sum[ Binomial[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])]. 0
1, 1, 1, 1, 17, 1, 1, 289, 289, 1, 1, 4913, 83521, 4913, 1, 1, 83521, 24137569, 24137569, 83521, 1, 1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1, 1, 24137569, 2015993900449, 582622237229761, 582622237229761, 2015993900449 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 19, 580, 93349, 48442182, 132542231095, 1169276510535560, 54387203223981636937, 8156589318954947565101962,

6449662938942774589680849369691,...}.

I put in a higher level one than Paul Hanna did instead of a comment.

I spent all morning looking for a q type polynomial that gave integer factorials.

This type of polynomial comes from the identity:

1/(1-x(^n=Sum[Binomial[n,i]*x^i,{i,0,Infinity}]

LINKS

Table of n, a(n) for n=0..33.

FORMULA

t(n,m)=If[m == 0, n!, Product[Sum[ Binomial[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, 17, 1},

{1, 289, 289, 1},

{1, 4913, 83521, 4913, 1},

{1, 83521, 24137569, 24137569, 83521, 1},

{1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1},

{1, 24137569, 2015993900449, 582622237229761, 582622237229761, 2015993900449, 24137569, 1},

{1, 410338673, 582622237229761, 2862423051509815793, 48661191875666868481, 2862423051509815793, 582622237229761, 410338673, 1},

{1, 6975757441, 168377826559400929, 14063084452067724991009, 4064231406647572522401601, 4064231406647572522401601, 14063084452067724991009, 168377826559400929, 6975757441, 1},

{1, 118587876497, 48661191875666868481, 69091933913008732880827217, 339448671314611904643504117121, 5770627412348402378939569991057, 339448671314611904643504117121, 69091933913008732880827217, 48661191875666868481, 118587876497, 1}

MATHEMATICA

Clear[t, n, m, i, k, a, b] t[n_, m_] = If[m == 0, n!, Product[Sum[Binomial[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

A118180, A118185, A118190

Sequence in context: A176794 A176244 A022180 * A015143 A172196 A056110

Adjacent sequences:  A156578 A156579 A156580 * A156582 A156583 A156584

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 10 2009

STATUS

approved

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Last modified February 21 01:29 EST 2019. Contains 320364 sequences. (Running on oeis4.)