A254707 Expansion of (1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

1, 0, 4, 1, 8, 4, 15, 8, 25, 15, 38, 25, 55, 38, 77, 55, 103, 77, 135, 103, 173, 135, 217, 173, 268, 217, 327, 268, 393, 327, 468, 393, 552, 468, 645, 552, 748, 645, 862, 748, 986, 862, 1122, 986, 1270, 1122, 1430, 1270, 1603, 1430, 1790, 1603, 1990, 1790

Offset

0,3

Comments

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.

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

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).

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.

a(n+3) - a(n) = 0 if n even else A006578((n+5)/2) for all n in Z.

a(n+2) = 2*A254594(n) + A254594(n+2) for all n in Z.

a(n) = -A254708(-9 - n) for all n in Z.

Example

G.f. = 1 + 4*x^2 + x^3 + 8*x^4 + 4*x^5 + 15*x^6 + 8*x^7 + 25*x^8 + ...

Mathematica

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 8 n^2 + 9 n + 18, 17 n^2 + 84 n + 148], 96];
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}]];
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];

Prog

(PARI) {a(n) = (n^3 + if( n%2, 8*n^2 + 9*n + 18, 17*n^2 + 84*n + 148)) \ 96};
(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)};

Crossrefs

Cf. A006578, A254594, A254708.

Sequence in context: A021709 A193337 A141567 * A134417 A116080 A343125

Adjacent sequences: A254704 A254705 A254706 * A254708 A254709 A254710

Keyword

nonn,easy

Author

Michael Somos, Feb 06 2015

Status

approved