A131606 Triangle read by rows: row n gives coefficients of the polynomial p(x, n) = Sum[Fibonacci[n]^i*x^(n - i), {i, 0, n}].
1, 1, 1, 1, 1, 1, 8, 4, 2, 1, 81, 27, 9, 3, 1, 3125, 625, 125, 25, 5, 1, 262144, 32768, 4096, 512, 64, 8, 1, 62748517, 4826809, 371293, 28561, 2197, 169, 13, 1, 37822859361, 1801088541, 85766121, 4084101, 194481, 9261, 441, 21, 1, 60716992766464
Offset: 0
Examples
Triangle begins:
{1},
{1, 1},
{1, 1, 1},
{8, 4, 2, 1},
{81, 27, 9, 3, 1},
{3125, 625, 125, 25, 5, 1},
{262144, 32768, 4096, 512, 64, 8, 1},
{62748517, 4826809, 371293, 28561, 2197, 169, 13, 1},
{37822859361, 1801088541, 85766121, 4084101, 194481, 9261, 441, 21, 1},
{60716992766464, 1785793904896, 52523350144, 1544804416, 45435424, 1336336, 39304, 1156, 34, 1},
{253295162119140625, 4605366583984375, 83733937890625, 1522435234375, 27680640625, 503284375, 9150625, 166375, 3025, 55, 1}
Programs
-
Mathematica
Clear[p, a] a[n_] = Fibonacci[n]; p[x, 0] = 1; p[x_, n_] := p[x, n] = Sum[a[n]^i*x^(n - i), {i, 0, n}]; Table[p[x, n], {n, 0, 10}]; a0 = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a0] Table[Apply[Plus, CoefficientList[p[x, n], x]], {n, 0, 10}]
Extensions
Edited by N. J. A. Sloane, May 27 2008
Comments