A060345 An expansion related to Yukawa coupling.

5, 2875, 4876875, 8564575000, 15517926796875, 28663236110956000, 53621944306062201000, 101216230345800061125625, 192323666400003538944396875, 367299732093982242625847031250, 704288164978454714776724365580000, 1354842473951260627644461070753075500, 2613295702542192770504516764304958585000

Offset

0,1

Comments

Coefficients of 3-point function in dimension 3 [Morrison].

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..300

P. Candelas et al., A pair of Calabi-yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.

Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv:math/0610286 [math.NT], 2006.

David R. Morrison, Mathematical Aspects of Mirror Symmetry, arXiv:alg-geom/9609021, 1996, see Table 1 p. 60; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340.

Formula

Sum_{n >= 0} a(n)*q^n = 5 + Sum_{n >= 1} A060041(n)*n^3*q^n/(1-q^n).

Example

a(10) = A060041(1) + 8*A060041(2) + 125*A060041(5) + 1000*A060041(10) = 704288164978454714776724365580000.

Prog

(PARI) cumsum(v) = for(i=2, #v, v[i] += v[i-1]); v;
seq(N, {d=5}) = {
  my(x = 'x + O('x^(N+1)), h = cumsum(vector(d*N, n, 1/n)),
     y0 = sum(n=0, N, (d*n)!/n!^d * x^n),
     y1 = d * sum(n = 1, N, ((d*n)!/n!^d * (h[d*n] - h[n])) * x^n),
     Qx = x * exp(y1/y0), Xq = serreverse(Qx));
  Vec(d * (x * Xq'/Xq)^(d-2) / ((1 - d^d*Xq) * sqr(subst(y0, 'x, Xq))));
};
seq(20)  \\ Gheorghe Coserea, Jul 29 2016

Crossrefs

Cf. A060041, A076909-A076917, A076923.

Sequence in context: A060041 A199878 A076908 * A361450 A171980 A347607

Adjacent sequences: A060342 A060343 A060344 * A060346 A060347 A060348

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 30 2001

Extensions

More terms from Vladeta Jovovic, Apr 01 2001

a(6) corrected and a(10)-a(12) added by Gheorghe Coserea, Jul 28 2016

Status

approved