A199627 G.f.: (1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4)) for g=1.

1, 2, 1, 1, 2, 2, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111

Offset: 0

Author

N. J. A. Sloane, Nov 08 2011

Keywords

• nonn
• easy

Comments

Expansion of a Poincaré series [or Poincare series] for space of moduli M_2 of stable bundles.

Links

• Georg Fischer, Table of n, a(n) for n = 0..999
• Bott, Raoul, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), no. 2, 331-358; reprinted in Vol. 48 (October, 2011). See Eq. (4.30).
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A047538.

Programs

Magma

g:=1; m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4))));  // Bruno Berselli, Nov 08 2011

Maple

f:=g->(1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4));
s:=g->seriestolist(series(f(g),x,60));
s(1);

PARI

Vec((1 + x)^2*(1 - 2*x + 2*x^2 - x^3 - x^4 + 3*x^5 - 2*x^6 + x^7) / ((1 - x)^2*(1 + x^2)) + O(x^70)) \\ Colin Barker, Nov 05 2019

Formula

a(n) = A047538(n-3) for n >= 6. - Georg Fischer, Oct 28 2018

From Colin Barker, Nov 05 2019: (Start)

G.f.: (1 + x)^2*(1 - 2*x + 2*x^2 - x^3 - x^4 + 3*x^5 - 2*x^6 + x^7) / ((1 - x)^2*(1 + x^2)).

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>9.

a(n) = (-16 + (-i)^(1+n) + i^(1+n) + 4*n) / 2 for n>5, where i=sqrt(-1).

(End)