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%I #12 Mar 11 2021 05:21:01
%S 0,-1,1,-2,0,2,-7,-3,3,7,-20,-56,0,56,20,-61,-415,-370,370,415,61,
%T -182,-2632,-5710,0,5710,2632,182,-547,-15155,-64407,-49735,49735,
%U 64407,15155,547,-1640,-82896,-619696,-1085840,0,1085840,619696,82896,1640
%N Triangle of coefficients of p(x,n) = (1/4)*(1-x)^(n+1)*Sum_{m >= 0} ((2*m- 1)^n - (2*m+3)^n)*x^m, read by rows.
%C Row sums are zero.
%H G. C. Greubel, <a href="/A154852/b154852.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Rows are coefficients of p(x,n) = (1/4)*(1-x)^(n+1)*Sum_{m >= 0} ((2*m-1)^n - (2*m+3)^n)*x^m.
%e Triangle begins as:
%e 0;
%e -1, 1;
%e -2, 0, 2;
%e -7, -3, 3, 7;
%e -20, -56, 0, 56, 20;
%e -61, -415, -370, 370, 415, 61;
%e -182, -2632, -5710, 0, 5710, 2632, 182;
%e -547, -15155, -64407, -49735, 49735, 64407, 15155, 547;
%e -1640, -82896, -619696, -1085840, 0, 1085840, 619696, 82896, 1640;
%t T[n_,k_,p_,q_,r_,t_]:= SeriesCoefficient[(1/p)*(1-x)^(n+1)*Sum[((q*j+r)^n - (q*j+t)^n)*x^j, {j, 0, n}], {x,0,k}];
%t Table[T[n,k,4,2,-1,3], {n,0,12},{k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 11 2021 *)
%o (Sage)
%o def f(n,p,q,r,t,x) : return (1/p)*(1-x)^(n+1)*sum( ((q*j+r)^n - (q*j+t)^n )*x^j for j in (0..n))
%o [[( f(n,4,2,-1,3,x) ).series(x,n+1).list()[k] for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Mar 11 2021
%Y Cf. A154853, A154854, A154855.
%K tabl,sign
%O 0,4
%A _Roger L. Bagula_, Jan 16 2009
%E Edited by _G. C. Greubel_, Mar 11 2021
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