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%I #15 May 18 2019 02:33:03
%S 0,3,-1,-3,4,-1,10,-6,3,-1,-18,14,-4,3,-1,42,-24,10,-5,3,-1,-87,47,
%T -19,12,-5,3,-1,190,-83,42,-22,11,-5,3,-1,-405,152,-84,45,-20,11,-5,3,
%U -1
%N Array T(n,m) read by antidiagonals: the coefficient of [x^m] of 1/(-x^n + 1 - x^(1+n) + x + 3*x^(2+n) - 2*x^2 - x^(3+n)) in row n, column 1 <= m.
%C Antidiagonal sums are: 0, 2, 0, 6, -6, 25, -50, 135, -304, ...
%C For large n, the rows approach A077925.
%F G.f.: 1/( - x^n + 1 - x^(1+n) + x + 3*x^(2+n) - 2*x^2 - x^(3+n) ).
%e The array starts in row n=1 with columns m >= 1 as
%e 0, 3, -3, 10, -18, 42, -87, 190, -405, 873, ... A077899
%e -1, 4, -6, 14, -24, 47, -83, 152, -268, 476, ... A175722
%e -1, 3, -4, 10, -19, 42, -84, 174, -353, 726, ...
%e -1, 3, -5, 12, -22, 45, -87, 174, -340, 670, ...
%e -1, 3, -5, 11, -20, 42, -83, 169, -339, 686, ...
%e -1, 3, -5, 11, -21, 44, -86, 173, -343, 685, ...
%e -1, 3, -5, 11, -21, 43, -84, 170, -339, 681, ...
%e -1, 3, -5, 11, -21, 43, -85, 172, -342, 685, ...
%e -1, 3, -5, 11, -21, 43, -85, 171, -340, 682, ...
%e -1, 3, -5, 11, -21, 43, -85, 171, -341, 684, ...
%p A175721 := proc(m,k) 1/(-x^m + 1 - x^(1 + m) + x + 3*x^(2 + m) - 2* x^2 - x^(3 + m)) ; coeftayl(%,x=0,k) ; end proc: # _R. J. Mathar_, Dec 22 2010
%t f[x_, m_] = ExpandAll[(x - x^(m + 1))*(1 - x - x^2) - (1 - 2*x + x^(m + 1))];
%t g[x_, n_] = ExpandAll[x^(m + 3)*f[1/x, m]];
%t a = Table[Table[SeriesCoefficient[
%t Series[1/g[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
%t Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}]
%t Flatten[%]
%Y Cf. A000045, A077899, A077925, A175722.
%K sign,tabl
%O 1,2
%A _Roger L. Bagula_, Dec 04 2010
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