A133333 Irregular triangle read by rows: coefficients of Olinde Rodrigues recursive polynomial for inversions of permutations applied to Bonnaci type polynomials: x - 1, x^2 - x - 1, x^3 - x^2 - x - 1, etc.

-1, 1, -1, 3, -3, 1, 1, 4, 2, -8, -5, 8, 2, -4, 1, -1, -5, -15, -25, -25, -1, 25, 35, 5, -15, -21, 5, 5, 5, -5, 1, 1, 6, 21, 56, 114, 186, 246, 246, 171, 34, -114, -174, -149, -54, 54, 66, 51, 6, -34, -6, -6, 4, 9, -6, 1, -1, -7, -28, -84, -210, -448, -833, -1373, -2023, -2653, -3094, -3178, -2793, -1953, -883, 161, 917, 1197

Offset

1,4

Comments

The polynomial powers grow as I(n) = n!*binomial(n,2)/2.

Links

Table of n, a(n) for n=1..74.

Warren P. Johnson, Mathematics and Social Utopias in France: Olinde Rodrigues and His times by Simon Altmann; Eduardo L. Ortiz, American Math. Monthly, Oct 2007, volume 114, number 8, pages 752-758.

Example

{-1},
{1},
{-1, 3, -3, 1},
{1, 4, 2, -8, -5, 8, 2, -4, 1},
{-1, -5, -15, -25, -25, -1, 25, 35, 5, -15, -21, 5, 5, 5, -5, 1},
{1, 6, 21, 56, 114, 186, 246, 246, 171, 34, -114, -174, -149, -54, 54, 66, 51, 6, -34, -6, -6, 4, 9, -6,1},
{-1, -7, -28, -84, -210, -448, -833, -1373, -2023, -2653, -3094, -3178, -2793, -1953, -883, 161, 917, 1197, 987, 567,91, -253, -343, -203, -98, 28, 91, 63, -15, 7, -14, -14, 0, 14, -7, 1},

Mathematica

f[q_, n_] = If[n == 0, -1, q^(n - 1) - Sum[q^i, {i, 0, n - 2}]]; g[q_, n_] = Product[f[q, n], {m, 0, n}]; a = Table[CoefficientList[g[x, n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A197928 A109439 A247646 * A296523 A171876 A306462

Adjacent sequences: A133330 A133331 A133332 * A133334 A133335 A133336

Keyword

uned,sign,tabf

Author

Roger L. Bagula, Oct 19 2007

Status

approved