A251260 - OEIS

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

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OFFSET

0,2

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).

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)).

EXAMPLE

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

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];

PROG

(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)};

(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

CROSSREFS

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

Sequence in context: A101136 A036957 A319390 * A022943 A068491 A364796

Adjacent sequences: A251257 A251258 A251259 * A251261 A251262 A251263

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Mar 20 2015

STATUS

approved