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%I #5 Aug 09 2015 01:42:59
%S 1,1,5,1,1,14,36,14,1,1,27,169,321,169,27,1,1,44,496,2024,3268,2024,
%T 496,44,1,1,65,1145,7930,24740,36244,24740,7930,1145,65,1,1,90,2276,
%U 23750,119393,310036,426128,310036,119393,23750,2276,90,1,1,119,4081,59619
%N Alternate terms of A001263 as polynomials divided by x+1 to give a new triangle of coefficients of even powered polynomials.
%C Row sums are:
%C Table[Apply[Plus, CoefficientList[Factor[a[[n]]]/(x + 1), x]], {n, 2, Length[a], 2}];
%C {1, 7, 66, 715, 8398, 104006, 1337220, 17678835, 238819350, 3282060210}.
%C This sequence was found while looking into _Gary W. Adamson_'s comment on A001263.
%F 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}
%e {1},
%e {1, 5, 1},
%e {1, 14, 36, 14, 1},
%e {1, 27, 169, 321, 169, 27, 1},
%e {1, 44, 496, 2024, 3268, 2024, 496, 44, 1},
%e {1, 65, 1145, 7930, 24740, 36244, 24740, 7930, 1145, 65, 1},
%e {1, 90, 2276, 23750, 119393, 310036, 426128, 310036, 119393, 23750, 2276, 90,1},
%e {1, 119, 4081, 59619, 437241, 1748943, 3976777, 5225273, 3976777, 1748943,437241, 59619, 4081, 119, 1},
%e {1, 152, 6784, 131936, 1324624, 7511840, 25309312, 52054832, 66140388, 52054832, 25309312, 7511840, 1324624, 131936, 6784, 152, 1},
%e {1, 189, 10641, 265524, 3490320, 26556432, 123677328, 364582392, 693313668, 858267220, 693313668, 364582392, 123677328, 26556432, 3490320, 265524, 10641, 189, 1}
%t 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]
%Y Cf. A001263.
%K nonn,uned,tabf
%O 1,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Mar 18 2008
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