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A060345 An expansion related to Yukawa coupling.

%I A060345 #38 May 14 2020 05:13:26
%S A060345 5,2875,4876875,8564575000,15517926796875,28663236110956000,
%T A060345 53621944306062201000,101216230345800061125625,
%U A060345 192323666400003538944396875,367299732093982242625847031250,704288164978454714776724365580000,1354842473951260627644461070753075500,2613295702542192770504516764304958585000
%N A060345 An expansion related to Yukawa coupling.
%C A060345 Coefficients of 3-point function in dimension 3 [Morrison].
%H A060345 Gheorghe Coserea, <a href="/A060345/b060345.txt">Table of n, a(n) for n = 0..300</a>
%H A060345 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 A060345 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 A060345 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 A060345 Sum_{n >= 0} a(n)*q^n = 5 + Sum_{n >= 1} A060041(n)*n^3*q^n/(1-q^n).
%e A060345 a(10) = A060041(1) + 8*A060041(2) + 125*A060041(5) + 1000*A060041(10) = 704288164978454714776724365580000.
%o A060345 (PARI) cumsum(v) = for(i=2, #v, v[i] += v[i-1]); v;
%o A060345 seq(N, {d=5}) = {
%o A060345   my(x = 'x + O('x^(N+1)), h = cumsum(vector(d*N, n, 1/n)),
%o A060345      y0 = sum(n=0, N, (d*n)!/n!^d * x^n),
%o A060345      y1 = d * sum(n = 1, N, ((d*n)!/n!^d * (h[d*n] - h[n])) * x^n),
%o A060345      Qx = x * exp(y1/y0), Xq = serreverse(Qx));
%o A060345   Vec(d * (x * Xq'/Xq)^(d-2) / ((1 - d^d*Xq) * sqr(subst(y0, 'x, Xq))));
%o A060345 };
%o A060345 seq(20)  \\ _Gheorghe Coserea_, Jul 29 2016
%Y A060345 Cf. A060041, A076909-A076917, A076923.
%K A060345 nonn,easy
%O A060345 0,1
%A A060345 _N. J. A. Sloane_, Mar 30 2001
%E A060345 More terms from _Vladeta Jovovic_, Apr 01 2001
%E A060345 a(6) corrected and a(10)-a(12) added by _Gheorghe Coserea_, Jul 28 2016