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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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).
LINKS
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.
EXAMPLE
G.f. = 2 + 5*x^2 + 2*x^3 + 10*x^4 + 5*x^5 + 18*x^6 + 10*x^7 + 29*x^8 + ...
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 *)
PROG
(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)};
(Magma) m:=50; R<x>:=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
CROSSREFS
Sequence in context: A351481 A014842 A360559 * A132816 A261068 A077453
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Feb 06 2015
STATUS
approved

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Last modified June 16 11:07 EDT 2024. Contains 373429 sequences. (Running on oeis4.)