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

Author

Michael Somos, Feb 06 2015

Keywords

• nonn
• easy

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.

Examples

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

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).

Crossrefs

Cf. A006578, A254594, A254708.

Programs

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];

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)};

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.