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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. 1
1, 1, 3, 4, 8, 9, 16, 18, 28, 31, 45, 49, 68, 73, 97, 104, 134, 142, 179, 189, 233, 245, 297, 311, 372, 388, 458, 477, 557, 578, 669, 693, 795, 822, 936, 966, 1093, 1126, 1266, 1303, 1457, 1497, 1666, 1710, 1894, 1942, 2142, 2194, 2411, 2467, 2701, 2762, 3014 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

FORMULA

Euler transform of length 5 sequence [ 1, 2, 1, 1, -1].

a(n) = -a(-7-n) for all n in Z.

a(2*n) = A254875(n), a(2*n + 1) = A254874(n).

EXAMPLE

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 + ...

MATHEMATICA

a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 48 n^2 + 141 n + 162, 57 n^2 + 204 n + 288], 288];

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}]];

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];

PROG

(PARI) {a(n) = (5*n^3 + if( n%2, 48*n^2 + 141*n + 162, 57*n^2 + 204*n + 288 )) \ 288};

(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)};

CROSSREFS

Cf. A254874, A254875.

Sequence in context: A217788 A273257 A249485 * A193351 A243985 A200727

Adjacent sequences:  A254874 A254875 A254876 * A254878 A254879 A254880

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Feb 09 2015

STATUS

approved

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Last modified February 28 23:19 EST 2020. Contains 332353 sequences. (Running on oeis4.)