%I #2 Oct 12 2012 14:54:51

%S 1,2,4,1,6,24,22,8,1,24,144,260,176,62,12,1,120,960,2640,3120,1846,

%T 616,126,16,1,720,7200,26760,47040,43352,23376,7772,1632,222,20,1,

%U 5040,60480,283920,672000,882336,692160,347152,115680,25806,3800,366,24,1

%N A PolyLog functional polynomial coefficient triangular sequence: p(x,n)=(-1)^(n + 1)*(4*x + x^2)^(n + 1)*PolyLog[ -n, 1 + 4*x + x^2]/(1 + 4*x + x^2).

%C Row sums are:

%C {1, 7, 61, 679, 9445, 158095, 3088765, 68958295, 1731875605, 48328686175}.

%F p(x,n)=(-1)^(n + 1)*(4*x + x^2)^(n + 1)*PolyLog[ -n, 1 + 4*x + x^2]/(1 + 4*x + x^2); t(n,m)=corefficients(p(x,n)).

%e {1},

%e {2, 4, 1},

%e {6, 24, 22, 8, 1},

%e {24, 144, 260, 176, 62, 12, 1},

%e {120, 960, 2640, 3120, 1846, 616, 126, 16, 1},

%e {720, 7200, 26760, 47040, 43352, 23376, 7772, 1632, 222, 20, 1},

%e {5040, 60480, 283920, 672000, 882336, 692160, 347152, 115680, 25806, 3800, 366, 24, 1},

%e {40320, 564480, 3205440, 9596160, 16690464, 17898048, 12504368, 5939584, 1972216, 462544, 75716, 8336, 590, 28, 1},

%e {362880, 5806080, 38707200, 140555520, 307308960, 427351680, 395017440, 252562560, 115055176, 38059712, 9226768, 1635344, 207430, 17864, 958, 32, 1},

%e {3628800, 65318400, 500169600, 2138572800, 5660202240, 9788567040, 11491935360, 9475050240, 5648068592, 2487777248, 820929560, 204235072, 38286104, 5359312, 546092, 38080, 1598, 36, 1}

%t Clear[p, x, n, m]; p[x_, n_] = (-1)^(n + 1)*(4*x + x^2)^(n + 1)*PolyLog[ -n, 1 + 4*x + x^2]/(1 + 4*x + x^2); Table[FullSimplify[Expand[p[x, n]]], {n, 1, 10}]; Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 1, 10}]; Flatten[%]

%Y Cf. A008292, A123125.

%K nonn,uned

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008