A098494 Triangle read by rows: coefficients of polynomials E(n,x) related to partitions with parts occurring at most thrice.

1, 1, -1, 1, -5, 4, 1, -12, 35, -30, 1, -22, 143, -362, 312, 1, -35, 405, -2065, 4814, -4200, 1, -51, 925, -7965, 35434, -78744, 69120, 1, -70, 1834, -24010, 173929, -709240, 1525236, -1345680, 1, -92, 3290, -61040, 655529, -4235588, 16255420, -34148400, 30240000

Offset

0,5

Comments

The polynomials generate (-1)^k*n! times the diagonals of A098493.

Links

Seiichi Manyama, Rows n = 0..139, flattened

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Formula

E(n+1,x+1) - E(n+1,x) = (n+1) * ( E(n,x) - n * E(n-1,x-1) ).

Example

E(0,x) = 1
E(1,x) = x - 1
E(2,x) = x^2 - 5*x + 4
E(3,x) = x^3 - 12*x^2 + 35*x - 30
E(4,x) = x^4 - 22*x^3 + 143*x^2 - 362*x + 312
E(5,x) = x^5 - 35*x^4 + 405*x^3 - 2065*x^2 + 4814*x - 4200

Crossrefs

Columns include -A000326.

Constant terms E(n, 0) = -E(n-1, -1) = n!/2*A085455 = (-1)^n*n!*A005773.

Row sums are E(n, 1) = (-1)^n*n!*A005774(n-2). [corrected by Seiichi Manyama, Feb 04 2023]

Sequence in context: A005752 A364355 A365465 * A008955 A182824 A329120

Adjacent sequences: A098491 A098492 A098493 * A098495 A098496 A098497

Keyword

sign,tabl

Author

Ralf Stephan, Sep 12 2004

Status

approved