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}].

1, -1, 3, 9, 6, 9, -73, -75, 9, 27, 849, 1644, 774, 108, 81, -12241, -33849, -28098, -6426, 243, 243, 211929, 763314, 938007, 442044, 60183, 1458, 729, -4280473, -18995271, -31035393, -22471479, -6681123, -528525, 3645, 2187, 98806689, 521068632, 1064559708, 1049509224, 501783174, 99717480, 4802652, 17496, 6561

Offset: 0

Author

Roger L. Bagula, Jan 12 2009

Keywords

• uned
• tabl
• sign

Comments

Row sums are: {1, 2, 24, -112, 3456, -80128, 2417664, -83986432, 3340271616, -149428830208, 7427651272704,...}

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

Examples

			Triangle begins:
  {1},
  {-1, 3},
  {9, 6, 9},
  {-73, -75, 9, 27},
  {849, 1644, 774, 108, 81},
  {-12241, -33849, -28098, -6426, 243, 243},
  {211929, 763314, 938007, 442044, 60183, 1458, 729},
  {-4280473, -18995271, -31035393, -22471479, -6681123, -528525, 3645, 2187},
  {98806689, 521068632, 1064559708, 1049509224, 501783174, 99717480, 4802652, 17496, 6561},
  ...
		

Programs

Mathematica

Clear[p, a, b, c, d, n];
{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}];
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Formula

{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}];

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

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