|
%I #4 Mar 30 2012 17:34:27
%S 1,1,1,1,3,1,1,4,4,1,1,5,10,5,1,1,6,21,21,6,1,1,7,41,86,41,7,1,1,8,78,
%T 337,337,78,8,1,1,9,148,1247,2486,1247,148,9,1,1,10,283,4377,15745,
%U 15745,4377,283,10,1,1,11,547,14728,88444,156442,88444,14728,547,11,1,1
%N A Pascal triangle with an Eulerian-number shift: p(x,n)=If[n < 1, (x + 1)^(n + 1), (x + 1)^(n + 1) + (1 - x)^(n + 1)*PolyLog[ -n, x]].
%C Row sums are {1, 2, 5, 10, 22, 56, 184, 848, 5296, 40832, 363904, 3630848, ...}.
%F p(x,n)=If[n < 1, (x + 1)^(n + 1), (x + 1)^(n + 1) + (1 - x)^(n + 1)*PolyLog[ -n, x]]; t(n,m)=Coefficients(p(x,n)0.
%e {1}, {1, 1}, {1, 3, 1}, {1, 4, 4, 1}, {1, 5, 10, 5, 1}, {1, 6, 21, 21, 6, 1}, \ {1, 7, 41, 86, 41, 7, 1}, {1, 8, 78, 337, 337, 78, 8, 1}, {1, 9, 148, 1247, 2486, 1247, 148, 9, 1}, {1, 10, 283, 4377, 15745, 15745, 4377, 283, 10, 1}, {1, 11, 547, 14728, 88444, 156442, 88444, 14728, 547, 11, 1}, {1, 12, 1068, 48005, 455522, 1310816, 1310816, 455522, 48005, 1068, 12, 1}
%t Clear[p, n, m]; p[x_, n_] = If[n < 1, (x + 1)^(n + 1), (x + 1)^(n + 1) + (1 - x)^(n + 1)*PolyLog[ -n, x]]; Table[FullSimplify[ExpandAll[p[x, n]]], {n, -1, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, -1, 10}]; Flatten[%]
%K nonn,tabl
%O -1,5
%A _Roger L. Bagula_, Nov 04 2008
%E Edited by _N. J. A. Sloane_, Nov 07 2008
|
