A060345 - OEIS

%I #38 May 14 2020 05:13:26

%S 5,2875,4876875,8564575000,15517926796875,28663236110956000,

%T 53621944306062201000,101216230345800061125625,

%U 192323666400003538944396875,367299732093982242625847031250,704288164978454714776724365580000,1354842473951260627644461070753075500,2613295702542192770504516764304958585000

%N An expansion related to Yukawa coupling.

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

%H Gheorghe Coserea, <a href="/A060345/b060345.txt">Table of n, a(n) for n = 0..300</a>

%H P. Candelas et al., <a href="http://dx.doi.org/10.1016/0550-3213(91)90292-6">A pair of Calabi-yau manifolds as an exactly soluble superconformal theory</a>, Nuclear Phys. B 359 (1991), 21-74.

%H Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, <a href="https://arxiv.org/abs/math/0610286">Sequences of enumerative geometry: congruences and asymptotics</a>, arXiv:math/0610286 [math.NT], 2006.

%H David R. Morrison, <a href="https://arxiv.org/abs/alg-geom/9609021">Mathematical Aspects of Mirror Symmetry</a>, 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.

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

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

%o (PARI) cumsum(v) = for(i=2, #v, v[i] += v[i-1]); v;

%o seq(N, {d=5}) = {

%o   my(x = 'x + O('x^(N+1)), h = cumsum(vector(d*N, n, 1/n)),

%o      y0 = sum(n=0, N, (d*n)!/n!^d * x^n),

%o      y1 = d * sum(n = 1, N, ((d*n)!/n!^d * (h[d*n] - h[n])) * x^n),

%o      Qx = x * exp(y1/y0), Xq = serreverse(Qx));

%o   Vec(d * (x * Xq'/Xq)^(d-2) / ((1 - d^d*Xq) * sqr(subst(y0, 'x, Xq))));

%o };

%o seq(20)  \\ _Gheorghe Coserea_, Jul 29 2016

%Y Cf. A060041, A076909-A076917, A076923.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_, Mar 30 2001

%E More terms from _Vladeta Jovovic_, Apr 01 2001

%E a(6) corrected and a(10)-a(12) added by _Gheorghe Coserea_, Jul 28 2016