cp's OEIS Frontend

A154593 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}].

%I A154593 #6 May 08 2025 22:55:47
%S A154593 1,-1,3,9,6,9,-73,-75,9,27,849,1644,774,108,81,-12241,-33849,-28098,
%T A154593 -6426,243,243,211929,763314,938007,442044,60183,1458,729,-4280473,
%U A154593 -18995271,-31035393,-22471479,-6681123,-528525,3645,2187,98806689,521068632,1064559708,1049509224,501783174,99717480,4802652,17496,6561
%N A154593 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 A154593 Row sums are: {1, 2, 24, -112, 3456, -80128, 2417664, -83986432, 3340271616, -149428830208, 7427651272704,...}
%C A154593 This result is from a scan of {a,b,c,d} that are quadratic symmetric.
%F A154593 {a, b, c, d} = {2, 3, 3, 2};
%F A154593 p(x,n)=(-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];
%F A154593 t(n,m)=coefficients(p(x,n)).
%F A154593 p(x,n)=(-2)^n *(-1 - 3 x)^(1 + n)* LerchPhi[2 + 3 x, -n, 3/2]
%e A154593 Triangle begins:
%e A154593   {1},
%e A154593   {-1, 3},
%e A154593   {9, 6, 9},
%e A154593   {-73, -75, 9, 27},
%e A154593   {849, 1644, 774, 108, 81},
%e A154593   {-12241, -33849, -28098, -6426, 243, 243},
%e A154593   {211929, 763314, 938007, 442044, 60183, 1458, 729},
%e A154593   {-4280473, -18995271, -31035393, -22471479, -6681123, -528525, 3645, 2187},
%e A154593   {98806689, 521068632, 1064559708, 1049509224, 501783174, 99717480, 4802652, 17496, 6561},
%e A154593   ...
%t A154593 Clear[p, a, b, c, d, n];
%t A154593 {a, b, c, d} = {2, 3, 3, 2};
%t A154593 p[x_, n_] = (-1)^(n)*(1 - d - c x)^(n + 1)*Sum[(a*k + b)^n*(c*x + d)^k, {k, 0, Infinity}];
%t A154593 Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
%t A154593 Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
%t A154593 Flatten[%]
%K A154593 uned,tabl,sign
%O A154593 0,3
%A A154593 _Roger L. Bagula_, Jan 12 2009