A167883 Triangle read by rows: row n gives coefficients of expansion of polynomial p(k,n) in powers of k, defined by p(k, 0) = 1, p(k, 1) = 1+2*k; for n>1, p(k,n) = If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))].

1, 1, 2, 1, 10, 10, 1, 32, 90, 60, 1, 74, 594, 1040, 520, 1, 224, 2226, 6684, 7800, 3120, 1, 352, 12124, 95304, 227052, 215280, 71760, 1, 1058, 38484, 358656, 1252980, 2008152, 1506960, 430560, 1, 1348, 142264, 4028712, 32909556, 97352640

Offset

0,3

Comments

Row sums are {1, 3, 21, 183, 2229, 20055, 621873, 5596851, 374989401, 3374904603, 422613054909, ...}

The remarkable thing about these polynomials is that their infinite sums are a symmetrical triangle.

The old definition was "Coefficients of a recursive polynomial:p(k,n)=If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))] ( correction with cubic term in the infinite sum)."

Links

G. C. Greubel, Table of n, a(n) for the first 40 rows

Formula

p(k,0) = 1; p(k,1) = 1+2*k; p(k,n) = If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))].

Example

The triangle begins:
{1},
{1, 2},
{1, 10, 10},
{1, 32, 90, 60},
{1, 74, 594, 1040, 520},
{1, 224, 2226, 6684, 7800, 3120},
{1, 352, 12124, 95304, 227052, 215280, 71760},
{1, 1058, 38484, 358656, 1252980, 2008152, 1506960, 430560},
{1, 1348, 142264, 4028712, 32909556, 97352640, 132914880, 86112000, 21528000},
{1, 4046, 434880, 12939720, 122900940, 489515256, 982860480, 1055825280, 581256000, 129168000},
{1, 4598, 1184922, 92796080, 2442817180, 21051364536, 73606098792, 129668682240, 123157690560, 60493680000, 12098736000},
...
The infinite sum triangle is:
Table[Sum[p[k, n]*x^k, {k, 0, Infinity}], {n, 0, 10}];
{1},
{1, 1},
{-1, -18, -1},
{1, 179, 179, 1}
{-1, -2224, -8030, -2224, -1},
{1, 20049, 167150, 167150, 20049, 1},
{-1, -5596844, -145462469, -524422080, -360876091, -48653716, 1},
{1, 5596843, 194116185, 885298171, 885298171, 194116185, 5596843, 1},
{-1, -374989392, -25339790572, -207966886768, -400645626534, -207966886768, -25339790572, -374989392, -1}

Mathematica

p[k, 0] := 1; p[k, 1] := 1 + 2*k; p[k_, n_] := If[Mod[n, 2] == 0, (1 + 2*k)*p[k, n - 1] + n*Binomial[n + 1, n - 1]*k*(k + 1)*p[k, n - 2], (1 + 2*k)*(1 + 3*(p[k, n - 1] - 1))];
Table[CoefficientList[ExpandAll[p[k, n]], k], {n, 0, 10}]; Flatten[%]

Crossrefs

Sequence in context: A244582 A279007 A163914 * A290596 A151504 A151507

Adjacent sequences: A167880 A167881 A167882 * A167884 A167885 A167886

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 14 2009

Extensions

Edited by G. C. Greubel and N. J. A. Sloane, Jul 01 2016

Status

approved