|
%I #25 Feb 04 2023 10:17:21
%S 1,1,-1,1,-5,4,1,-12,35,-30,1,-22,143,-362,312,1,-35,405,-2065,4814,
%T -4200,1,-51,925,-7965,35434,-78744,69120,1,-70,1834,-24010,173929,
%U -709240,1525236,-1345680,1,-92,3290,-61040,655529,-4235588,16255420,-34148400,30240000
%N Triangle read by rows: coefficients of polynomials E(n,x) related to partitions with parts occurring at most thrice.
%C The polynomials generate (-1)^k*n! times the diagonals of A098493.
%H Seiichi Manyama, <a href="/A098494/b098494.txt">Rows n = 0..139, flattened</a>
%H Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://cdm.ucalgary.ca/article/view/61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.
%F E(n+1,x+1) - E(n+1,x) = (n+1) * ( E(n,x) - n * E(n-1,x-1) ).
%e E(0,x) = 1
%e E(1,x) = x - 1
%e E(2,x) = x^2 - 5*x + 4
%e E(3,x) = x^3 - 12*x^2 + 35*x - 30
%e E(4,x) = x^4 - 22*x^3 + 143*x^2 - 362*x + 312
%e E(5,x) = x^5 - 35*x^4 + 405*x^3 - 2065*x^2 + 4814*x - 4200
%Y Columns include -A000326.
%Y Constant terms E(n, 0) = -E(n-1, -1) = n!/2*A085455 = (-1)^n*n!*A005773.
%Y Row sums are E(n, 1) = (-1)^n*n!*A005774(n-2). [corrected by _Seiichi Manyama_, Feb 04 2023]
%K sign,tabl
%O 0,5
%A _Ralf Stephan_, Sep 12 2004
|
