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A251260 Expansion of (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

%I A251260 #22 Jul 16 2025 13:26:07
%S A251260 1,2,3,6,8,13,16,24,28,40,45,61,68,89,97,124,134,167,179,219,233,281,
%T A251260 297,353,372,437,458,533,557,642,669,765,795,903,936,1056,1093,1226,
%U A251260 1266,1413,1457,1618,1666,1842,1894,2086,2142,2350,2411,2636,2701,2944
%N A251260 Expansion of (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.
%H A251260 G. C. Greubel, <a href="/A251260/b251260.txt">Table of n, a(n) for n = 0..2500</a>
%H A251260 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,0,-2,-2,0,1,2,0,-1).
%F A251260 a(n) = A165188(n+1) + A254708(n-1) = A254594(n-1) + A008763(n+4) for all n in Z.
%F A251260 0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
%F A251260 a(2*n) = A254875(n) for all n in Z.
%F A251260 G.f.: (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)).
%e A251260 G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 8*x^4 + 13*x^5 + 16*x^6 + 24*x^7 + ...
%t A251260 a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 66 n^2 + 249 n, 57 n^2 + 204 n] + 288, 288];
%t A251260 a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 6, (u + v < x + w && k == 0) || (u + v > x + w && x + u + v + w == 2 k + 1)}, {x, u, v, w, k}, Integers, 10^9];
%t A251260 LinearRecurrence[{0,2,1,0,-2,-2,0,1,2,0,-1},{1,2,3,6,8,13,16,24,28,40,45},60] (* _Harvey P. Dale_, Jul 16 2025 *)
%o A251260 (PARI) {a(n) = (5*n^3 + if( n%2, 66*n^2 + 249*n, 57*n^2 + 204*n) + 288) \ 288};
%o A251260 (PARI) {a(n) = polcoeff( if( n<0, n = -8-n; -(1 + x + 2*x^2 + x^3), 1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};
%o A251260 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 2*x+x^2+x^3)/((1-x^2)^2*(1-x^3)*(1-x^4)))); // _G. C. Greubel_, Aug 03 2018
%Y A251260 Cf. A008763, A165188, A254594, A254708, A254875.
%K A251260 nonn,easy
%O A251260 0,2
%A A251260 _Michael Somos_, Mar 20 2015