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A254877 Expansion of (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

%I A254877 #15 Jul 06 2017 01:28:17
%S A254877 1,1,3,4,8,9,16,18,28,31,45,49,68,73,97,104,134,142,179,189,233,245,
%T A254877 297,311,372,388,458,477,557,578,669,693,795,822,936,966,1093,1126,
%U A254877 1266,1303,1457,1497,1666,1710,1894,1942,2142,2194,2411,2467,2701,2762,3014
%N A254877 Expansion of (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.
%C A254877 The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, and either u+v <= x+w or x+u+v+w is even.
%H A254877 G. C. Greubel, <a href="/A254877/b254877.txt">Table of n, a(n) for n = 0..1000</a>
%H A254877 <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 A254877 Euler transform of length 5 sequence [ 1, 2, 1, 1, -1].
%F A254877 a(n) = -a(-7-n) for all n in Z.
%F A254877 a(2*n) = A254875(n), a(2*n + 1) = A254874(n).
%e A254877 G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 9*x^5 + 16*x^6 + 18*x^7 + 28*x^8 + ...
%t A254877 a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 48 n^2 + 141 n + 162, 57 n^2 + 204 n + 288], 288];
%t A254877 a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -7 - n; -1, 1] (1 - x^5)/((1 - x) (1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
%t A254877 a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, ((u + v <= x + w && x + u + v + w == 2 k + 1) || x + u + v + w == 2 k)}, {x, u, v, w, k}, Integers, 10^9];
%o A254877 (PARI) {a(n) = (5*n^3 + if( n%2, 48*n^2 + 141*n + 162, 57*n^2 + 204*n + 288 )) \ 288};
%o A254877 (PARI) {a(n) = my(s=(-1)^(n<0)); if( n<0, n = -7-n); s * polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};
%Y A254877 Cf. A254874, A254875.
%K A254877 nonn,easy
%O A254877 0,3
%A A254877 _Michael Somos_, Feb 09 2015