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%I #12 Jul 06 2017 01:28:22
%S 1,0,4,1,8,4,15,8,25,15,38,25,55,38,77,55,103,77,135,103,173,135,217,
%T 173,268,217,327,268,393,327,468,393,552,468,645,552,748,645,862,748,
%U 986,862,1122,986,1270,1122,1430,1270,1603,1430,1790,1603,1990,1790
%N Expansion of (1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.
%C The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+7 = x+u, u+v != x+w, and x+u+v+w is even.
%H G. C. Greubel, <a href="/A254707/b254707.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,0,-2,-2,0,1,2,0,-1).
%F G.f.: (1 + 2*x^2) / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
%F 0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - 2*a(n+4) + 2*a(n+6) + 2*a(n+7) + a(n+8) - a(n+9) - a(n+10) + 3 for all n in Z.
%F a(n+3) - a(n) = 0 if n even else A006578((n+5)/2) for all n in Z.
%F a(n+2) = 2*A254594(n) + A254594(n+2) for all n in Z.
%F a(n) = -A254708(-9 - n) for all n in Z.
%e G.f. = 1 + 4*x^2 + x^3 + 8*x^4 + 4*x^5 + 15*x^6 + 8*x^7 + 25*x^8 + ...
%t a[ n_] := Quotient[ n^3 + If[ OddQ[n], 8 n^2 + 9 n + 18, 17 n^2 + 84 n + 148], 96];
%t a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -9 - n; -2 - x^2, 1 + 2 x^2] / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
%t a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 7, u + v != x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
%o (PARI) {a(n) = (n^3 + if( n%2, 8*n^2 + 9*n + 18, 17*n^2 + 84*n + 148)) \ 96};
%o (PARI) {a(n) = polcoeff( if( n<0, n = -9-n; -2 - x^2, 1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};
%Y Cf. A006578, A254594, A254708.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Feb 06 2015
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