A123670 Triangle read by rows: T(n,k) is the coefficient of x^k of the polynomial n(n-x)(n-2x)(n-3x)...(n-(n-1)x) (n>=1, 0<=k<=n-1).

1, 4, -2, 27, -27, 6, 256, -384, 176, -24, 3125, -6250, 4375, -1250, 120, 46656, -116640, 110160, -48600, 9864, -720, 823543, -2470629, 2941225, -1764735, 557032, -86436, 5040, 16777216, -58720256, 84410368, -64225280, 27725824, -6723584, 836352, -40320, 387420489, -1549681956

Offset

1,2

Comments

T(n,0) = n^n = A000312(n). Row sums yield the factorials (A000142).

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Example

Triangular sequence:
{1},
{4, -2},
{27, -27, 6},
{256, -384, 176, -24},
{3125, -6250, 4375, -1250, 120},
{46656, -116640, 110160, -48600,9864, -720},
{823543, -2470629, 2941225, -1764735, 557032, -86436, 5040},
{16777216, -58720256, 84410368, -64225280, 27725824, -6723584, 836352, -40320},
{387420489, -1549681956, 2611501074, -2410616376, 1325591001, -441450324, 86112396, -8876304, 362880},
{10000000000, -45000000000, 87000000000, -94500000000, 63273000000, -26932500000, 7236800000, -1172700000, 102657600, -3628800}

Maple

T:=(n, k)->coeff(product(n-j*x, j=0..n-1), x, k): for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
a123670_row := proc(n) local k; seq(coeff(expand((-1)^n*n^(n-k)* pochhammer(-x, n)), x, n-k), k=0..n-1) end: # Peter Luschny, Nov 28 2010

Mathematica

S3[n_, x_] = Product[(n - m*x), {m, 0, n - 1}] Table[ExpandAll[S3[n, x]], {n, 0, 10}] w2 = Table[CoefficientList[S3[n, x], x], {n, 1, 10}] Flatten[w2]

Crossrefs

Cf. A048994, A008276.

Cf. A000312, A000142.

Sequence in context: A140331 A095896 A286798 * A200032 A121667 A368767

Adjacent sequences: A123667 A123668 A123669 * A123671 A123672 A123673

Keyword

sign,tabl

Author

Gary W. Adamson and Roger L. Bagula, Nov 13 2006

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

Status

approved