A254708 Expansion of (1 + 2*x^2) / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11) in powers of x.

2, 0, 5, 2, 10, 5, 18, 10, 29, 18, 43, 29, 62, 43, 85, 62, 113, 85, 147, 113, 187, 147, 233, 187, 287, 233, 348, 287, 417, 348, 495, 417, 582, 495, 678, 582, 785, 678, 902, 785, 1030, 902, 1170, 1030, 1322, 1170, 1486, 1322, 1664, 1486, 1855, 1664, 2060, 1855

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) or (u+v > x+w and x+u+v+w is odd).

Examples

			G.f. = 2 + 5*x^2 + 2*x^3 + 10*x^4 + 5*x^5 + 18*x^6 + 10*x^7 + 29*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. A001859, A254707, A254594.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2 + x^2)/(1-2*x^2-x^3+2*x^5+2*x^6-x^8-2*x^9+x^11))); // G. C. Greubel, Aug 03 2018

Mathematica

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 10 n^2 + 21 n + 12, 19 n^2 + 108 n + 192], 96];
a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -9 - n; -1 - 2 x^2, 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) || (u + v > x + w && x + u + v + w == 2 k + 1)}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[(2 + x^2)/(1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11), {x,0,50}], x] (* G. C. Greubel, Apr 14 2017 *)
LinearRecurrence[{0,2,1,0,-2,-2,0,1,2,0,-1},{2,0,5,2,10,5,18,10,29,18,43},60] (* Harvey P. Dale, Mar 13 2023 *)

PARI

{a(n) = (n^3 + if( n%2, 10*n^2 + 21*n + 12, 19*n^2 + 108*n + 192)) \ 96};

PARI

{a(n) = polcoeff( if( n<0, n = -9-n; -1 - 2*x^2, 2 + x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

Formula

G.f.: (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 A001859((n+5)/2) for all n in Z.

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

a(n) = -A254707(-9 - n) for all n in Z.