A158451 Coefficient triangle of permutation-based polynomials: p(x,n)=Product[x - n!/(n - i)!, {i, 0, n}].

1, -1, 1, 1, -2, 1, -4, 8, -5, 1, 108, -180, 87, -16, 1, -27648, 39168, -12912, 1456, -65, 1, 86400000, -110880000, 26094000, -1650400, 36725, -326, 1, -4031078400000, 4893281280000, -896020704000, 34174310400, -357808500, 1324056, -1957, 1

Offset

0,5

Comments

Row sums are zero.

Links

Table of n, a(n) for n=0..35.

Formula

p(x,n)=Product[x - n!/(n - i)!, {i, 0, n}];

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

Example

{1},
{-1, 1},
{1, -2, 1},
{-4, 8, -5, 1},
{108, -180, 87, -16, 1},
{-27648, 39168, -12912, 1456, -65, 1},
{86400000, -110880000, 26094000, -1650400, 36725, -326, 1},
{-4031078400000, 4893281280000, -896020704000, 34174310400, -357808500, 1324056, -1957, 1},
{3319766398771200000, -3895456048081920000, 590749069282848000, -15141472491398400, 82175247585900, -122793194244, 64892443, -13700, 1},
{-55696437941726556979200000, 63865801383873651671040000, -8324224766165177597952000, 155357479788681048883200, -496492360412254617600, 336652684227299328, -62874268569280, 4153225952, -109601, 1},
{21577941222941856209168026828800000, -24327154556554879191902469488640000, 2789027842618623276064941539328000, -39897072340016669775798416179200, 82592651793766428522451353600, -29319586168244438225553408, 2208824096557097896128, -45835678198307232, 336412288521, -986410, 1},
{-215779412229418562091680268288000000000000000, 240106566785973441283582737776640000000000000, -24606057340816741519508963642572800000000000, 279301754985262494304252138094592000000000, -399053316051960464186506613145600000000, 82595583752383252966273908940800000, -2931980705065409393534302080000, 22088286801249177268512000, -45835711839536084100, 33641238716200, -9864101, 1 }

Mathematica

Clear[p, x, n, i];
p[x_, n_] = Product[x - n!/(n - i)!, {i, 0, n}];
Table[ExpandAll[p[x, n]], {n, 0, 10}];
Join[{{1}}, Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]];
Flatten[%]
Join[{1}, Table[Apply[Plus, CoefficientList[ExpandAll[p[x, n]], x]], {n, 0, 10}]];

Crossrefs

Sequence in context: A026204 A059146 A059148 * A257706 A118272 A112173

Adjacent sequences: A158448 A158449 A158450 * A158452 A158453 A158454

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Mar 19 2009

Status

approved