|
%I #14 Sep 13 2016 07:59:27
%S 1,1,1,1,8,1,1,35,35,1,1,126,394,126,1,1,417,3062,3062,417,1,1,1324,
%T 19895,44680,19895,1324,1,1,4111,117021,503827,503827,117021,4111,1,1,
%U 12602,648616,4882342,9193838,4882342,648616,12602,1,1,38333,3464840,42960752,137516234,137516234,42960752,3464840,38333,1
%N A triangular sequence of coefficients of polynomials: p(x,n) = (2*(x - 1)^n * (Sum_{k>=0} (((-1)^n*(2*k + 1)^(n - 1)))*x^k) - (x - 1)^(n + 1)*(Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^k)/x).
%C Row sums are: {1, 2, 10, 72, 648, 6960, 87120, 1249920, 20280960, 367960320, ...}.
%H G. C. Greubel, <a href="/A154335/b154335.txt">Table of n, a(n) for n = 0..1274</a>
%F p(x,n) = (2*(x - 1)^n * (Sum_{k>=0} (((-1)^n*(2*k + 1)^(n - 1)))*x^k) - (x - 1)^(n + 1)*(Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^k)/x).
%F Functional form:
%F p(x,n) = (2*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog(-n, x)/x).
%F t(n,m) = Coefficients(p(x,n)).
%e {1},
%e {1, 1},
%e {1, 8, 1},
%e {1, 35, 35, 1},
%e {1, 126, 394, 126, 1},
%e {1, 417, 3062, 3062, 417, 1},
%e {1, 1324, 19895, 44680, 19895, 1324, 1},
%e {1, 4111, 117021, 503827, 503827, 117021, 4111, 1},
%e {1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1},
%e {1, 38333, 3464840, 42960752, 137516234, 137516234, 42960752, 3464840, 38333, 1}
%t Clear[p, x, n]; p[x_, n_] = (2*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}] - (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0,Infinity}]/x);
%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
%t Flatten[%]
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Jan 07 2009
|
