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

1, 2, 3, 6, 8, 13, 16, 24, 28, 40, 45, 61, 68, 89, 97, 124, 134, 167, 179, 219, 233, 281, 297, 353, 372, 437, 458, 533, 557, 642, 669, 765, 795, 903, 936, 1056, 1093, 1226, 1266, 1413, 1457, 1618, 1666, 1842, 1894, 2086, 2142, 2350, 2411, 2636, 2701, 2944

Offset: 0

Author

Michael Somos, Mar 20 2015

Keywords

• nonn
• easy

Examples

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 8*x^4 + 13*x^5 + 16*x^6 + 24*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Crossrefs

Cf. A008763, A165188, A254594, A254708, A254875.

Programs

Magma

m:=50; R:=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

Mathematica

a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 66 n^2 + 249 n, 57 n^2 + 204 n] + 288, 288];
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];
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 *)

PARI

{a(n) = (5*n^3 + if( n%2, 66*n^2 + 249*n, 57*n^2 + 204*n) + 288) \ 288};

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)};

Formula

a(n) = A165188(n+1) + A254708(n-1) = A254594(n-1) + A008763(n+4) for all n in Z.

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.

a(2*n) = A254875(n) for all n in Z.

G.f.: (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)).