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Search: id:A136267
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| A136267 |
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Alternate terms of A001263 as polynomials divided by x+1 to give a new triangle of coefficients of even powered polynomials. |
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+0
1
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| 1, 1, 5, 1, 1, 14, 36, 14, 1, 1, 27, 169, 321, 169, 27, 1, 1, 44, 496, 2024, 3268, 2024, 496, 44, 1, 1, 65, 1145, 7930, 24740, 36244, 24740, 7930, 1145, 65, 1, 1, 90, 2276, 23750, 119393, 310036, 426128, 310036, 119393, 23750, 2276, 90, 1, 1, 119, 4081, 59619
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums are:
Table[Apply[Plus, CoefficientList[Factor[a[[n]]]/(x + 1), x]], {n, 2, Length[a], 2}];
{1, 7, 66, 715, 8398, 104006, 1337220, 17678835, 238819350, 3282060210}.
this sequence was found while looking into Gary Adamson's comment on A001263.
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FORMULA
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T(n,m) = Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m p(x,n)=Sum[t(n,m)^x^(m-1),{m,1,n}]/(x+1): {n,2,limit,skip one}
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EXAMPLE
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{1},
{1, 5, 1},
{1, 14, 36, 14, 1},
{1, 27, 169, 321, 169, 27, 1},
{1, 44, 496, 2024, 3268, 2024, 496, 44, 1},
{1, 65, 1145, 7930, 24740, 36244, 24740, 7930, 1145, 65, 1},
{1, 90, 2276, 23750, 119393, 310036, 426128, 310036, 119393, 23750, 2276, 90,1},
{1, 119, 4081, 59619, 437241, 1748943, 3976777, 5225273, 3976777, 1748943,437241, 59619, 4081, 119, 1},
{1, 152, 6784, 131936, 1324624, 7511840, 25309312, 52054832, 66140388, 52054832, 25309312, 7511840, 1324624, 131936, 6784, 152, 1},
{1, 189, 10641, 265524, 3490320, 26556432, 123677328, 364582392, 693313668, 858267220, 693313668, 364582392, 123677328, 26556432, 3490320, 265524, 10641, 189, 1}
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MATHEMATICA
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T[n_, m_] := Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m; a = Table[Apply[Plus, Table[T[n, m]*x^(m - 1), {m, 1, n}]], {n, 1, 20}]; Table[Factor[a[[n]]]/(x + 1), {n, 2, Length[a], 2}]; b = Table[CoefficientList[Factor[a[[n]]]/(x + 1), x], {n, 2, Length[a], 2}]; Flatten[b]
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CROSSREFS
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Cf. A001263.
Sequence in context: A119725 A111910 A008957 this_sequence A109960 A056940 A074060
Adjacent sequences: A136264 A136265 A136266 this_sequence A136268 A136269 A136270
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KEYWORD
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nonn,uned,tabf
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AUTHOR
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Roger L. Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Mar 18 2008
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