A154593 - OEIS

%I #3 Mar 30 2012 17:34:29

%S 1,-1,3,9,6,9,-73,-75,9,27,849,1644,774,108,81,-12241,-33849,-28098,

%T -6426,243,243,211929,763314,938007,442044,60183,1458,729,-4280473,

%U -18995271,-31035393,-22471479,-6681123,-528525,3645,2187,98806689

%N A triangle of polynomial coefficients:{a, b, c, d} = {2, 3, 3, 2}; p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}].

%C Row sums are:

%C {1, 2, 24, -112, 3456, -80128, 2417664, -83986432, 3340271616, -149428830208, \ 7427651272704,...}

%C This result is from a scan of {a,b,c,d} that are quadratic symmetric.

%F {a, b, c, d} = {2, 3, 3, 2};

%F p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];

%F t(n,m)=coefficients(p(x,n)).

%F p(x,n)=(-2)^n *(-1 - 3 x)^(1 + n)* LerchPhi[2 + 3 x, -n, 3/2]

%e {1},

%e {-1, 3},

%e {9, 6, 9},

%e {-73, -75, 9, 27},

%e {849, 1644, 774,108, 81},

%e {-12241, -33849, -28098, -6426, 243, 243},211929, 763314, 938007, 442044, 60183, 1458, 729},

%e {-4280473, -18995271, -31035393, -22471479, -6681123, -528525, 3645, 2187},

%e {98806689, 521068632, 1064559708, 1049509224, 501783174, 99717480, 4802652, 17496, 6561},

%e {-2565862561, -15676328181, -38479393620, -48196931796, -32188014414, -10798177878, -1481190948, -42996420, 45927, 19683},

%e {74035143849, 514213447902, 1474631881461, 2244768645096, 1936955084130, 932814106740, 227841626466, 22003372008, 387709173, 196830, 59049}

%t Clear[p, a, b, c, d, n];

%t {a, b, c, d} = {2, 3, 3, 2};

%t p[x_, n_] = (-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];

%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

%t Flatten[%]

%K uned,tabl,sign

%O 0,3

%A _Roger L. Bagula_, Jan 12 2009