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 A136267 Alternate terms of A001263 as polynomials divided by x+1 to give a new triangle of coefficients of even powered polynomials. 0
 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; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 W. Adamson's comment on A001263. LINKS FORMULA 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} EXAMPLE {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} MATHEMATICA 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] CROSSREFS Cf. A001263. Sequence in context: A144438 A157207 A008957 * A109960 A196019 A056940 Adjacent sequences: A136264 A136265 A136266 * A136268 A136269 A136270 KEYWORD nonn,uned,tabf AUTHOR Roger L. Bagula and Gary W. Adamson, Mar 18 2008 STATUS approved

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Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)