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A156576
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A triangular sequence of the anti-diagonal of the q squared like factorial: t(n,m)=If[m == 0, n!, Product[Sum[(i + 1)*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].
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1, 1, 1, 1, 1, 2, 1, 1, 5, 6, 1, 1, 7, 85, 24, 1, 1, 9, 238, 4165, 120, 1, 1, 11, 513, 33796, 537285, 720, 1, 1, 13, 946, 160569, 18486412, 172468485, 5040, 1, 1, 15, 1573, 554356, 255786417, 37065256060, 132628264965, 40320, 1, 1, 17, 2430, 1549405
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Row sums are:
{1, 2, 4, 13, 118, 4534, 572327, 191121467, 169949903708, 503243397794904,
7528543001953603041,...}.
The q squared binomials turn out rational.
Based on polynomials:
p(x,n) =(1-x)^2*Sum[(k + 1)*(x)^k, {k, 0, n - 1}];
where:
q(x,n)= Sum[(k + 1)*(x)^k, {k, 0, n - 1}];
is a count up polynomial.
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FORMULA
| t(n,m)=If[m == 0, n!, Product[Sum[(i + 1)*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]];
out_(n,m)=anti-diagonal(t(n,m))
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EXAMPLE
| {1},
{1, 1},
{1, 1, 2},
{1, 1, 5, 6},
{1, 1, 7, 85, 24},
{1, 1, 9, 238, 4165, 120},
{1, 1, 11, 513, 33796, 537285, 720},
{1, 1, 13, 946, 160569, 18486412, 172468485, 5040},
{1, 1, 15, 1573, 554356, 255786417, 37065256060, 132628264965, 40320},
{1, 1, 17, 2430, 1549405, 2057215116, 1979019508329, 263459840074480, 237802479082245, 362880},
{1, 1, 19, 3553, 3727620, 11566308325, 46207108720476, 72054121278750561, 6482165905192505920, 974276756799957765, 3628800}
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MATHEMATICA
| Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(i + 1)*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];
Flatten[%]
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CROSSREFS
| Sequence in context: A119687 A086856 A052916 * A176093 A092437 A064814
Adjacent sequences: A156573 A156574 A156575 * A156577 A156578 A156579
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 10 2009
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