|
%I #6 Apr 12 2013 01:42:09
%S 1,1,-1,-18,-42,-18,-1,1,115,1539,5065,5065,1539,115,1,-1,-612,-30369,
%T -359056,-1439038,-2255448,-1439038,-359056,-30369,-612,-1,1,3109,
%U 487944,16069256,177275075,808273143,1688579472,1688579472,808273143
%N A triangle sequence of polynomial coefficients: p(x,n)=(x - 1)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x.
%C Row sums are:
%C {2, -80, 13440, -5913600, 5381376000, -8782405632000, 23361198981120000,
%C -94566133475573760000, 553211880832106496000000, -4492080472356704747520000000,...}
%F p(x,n)=(x - 1)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x;
%F t(n,m)=Coefficients(p(x,n)).
%e {1, 1},
%e {-1, -18, -42, -18, -1},
%e {1,115, 1539, 5065, 5065, 1539, 115, 1},
%e {-1, -612, -30369, -359056, -1439038, -2255448, -1439038, -359056, -30369, -612, -1},
%e {1, 3109,487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256,487944, 3109, 1},
%e {-1, -15606, -7232832, -588609722, -15102054532, -159360510654, -796011579264, -2034786608786, -2770692409206, -2034786608786, -796011579264, -159360510654, -15102054532, -588609722, -7232832, -15606, -1},
%e {1, 78103, 103694985, 19568948247, 1065525448614, 23072731441362, 236032579067166, 1262043871882890, 3749020958436984, 6409344151561648, 6409344151561648, 3749020958436984, 1262043871882890, 236032579067166, 23072731441362, 1065525448614, 19568948247, 103694985, 78103, 1},
%e {-1, -390600, -1466023731, -619322458800, -67773276182575, -2802617455410216, -53645573041228725, -536366226569480256, -3023314553367761850, -10090695137544912400, -20563892762682272046, -26024562946121517600, -20563892762682272046, -10090695137544912400, -3023314553367761850, -536366226569480256, -53645573041228725, -2802617455410216, -67773276182575, -619322458800, -1466023731, -390600, -1},
%e {1, 1953097, 20606359662, 19105228968022, 4062046061251702, 306352064179097622, 10355782284092172382, 180449348295691590742, 1772697944064120724647, 10422061778244020252047, 38179816523099760064252, 89512894147925375525772, 136527354458904110040052, 136527354458904110040052, 89512894147925375525772, 38179816523099760064252, 10422061778244020252047, 1772697944064120724647, 180449348295691590742, 10355782284092172382, 306352064179097622, 4062046061251702, 19105228968022, 20606359662, 1953097, 1},
%e {-1, -9765594, -288951921066, -581527646706874, -235124431637251555, -31362739743718489620, -1800493681167699729940, -52143903931162149522580, -843767280698373213383505, -8174015411024678270125030, -49739731928304174504355510, -196644863528988544778420550, -517044213844787972256968395, -918147140681329965091033240, -1110785055014320687664653080, -918147140681329965091033240, -517044213844787972256968395, -196644863528988544778420550, -49739731928304174504355510, -8174015411024678270125030, -843767280698373213383505, -52143903931162149522580, -1800493681167699729940, -31362739743718489620, -235124431637251555, -581527646706874, -288951921066, -9765594, -1}
%t Clear[p, x, n]; p[x_, n_] = (x - 1)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x;
%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
%t Flatten[%]
%Y Cf. A000680.
%K sign,tabf
%O 1,4
%A _Roger L. Bagula_, Jan 06 2009
|
