A290009 -id:A290009 - cp's OEIS frontend

A290010 Coefficients in expansion of 691E_6^2E_12.

691, -631008, 220745952, -97839374976, 19569814631136, 4535334591238080, 303188772007707264, 10484562664742729472, 226123357637141848800, 3395301924051385082784, 38309171317486687195200, 343029821520447762390144, 2537867756328299498904192

Offset: 0

Seiichi Manyama, Jul 17 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A013973 (E_6), A029828 (691*E_12), A280869 (E_6^2).

Cf. A290009.

terms = 13;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
691*E6[x]^2*E12[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A010839 Expansion of Product_{k >= 1} (1-x^k)^48.

Original entry on oeis.org

1, -48, 1080, -15040, 143820, -985824, 4857920, -16295040, 28412910, 38671600, -424520544, 1268350272, -1211937160, -4306546080, 18293091840, -23522231424, -26299018683, 137218594320, -150999182320, -134713340160

Offset: 0

N. J. A. Sloane

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			1 - 48*x + 1080*x^2 - 15040*x^3 + 143820*x^4 - 985824*x^5 + 4857920*x^6 - 16295040*x^7 + ...

Morris Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Column k=48 of A286354.

Cf. A000203, A082558, A126581, A282330 (E_8^3), A282332 (E_6*E_8*E_10 = E4*E_10^2), A290009, A290010.

Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_4, E_6, E_8 ; E_6, E_8, E_10 ; E_8, E_10, E_12]. G.f. is -691*b(q)/(q^2*1728^2*250). - Seiichi Manyama, Jul 17 2017
a(n) = (A290010(n+2) - A290009(n+2) + 691*(A282330(n+2) - A282332(n+2)))/(1728^2*250). - Seiichi Manyama, Jul 19 2017
a(0) = 1, a(n) = -(48/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

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A290010 Coefficients in expansion of 691E_6^2E_12.

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A010839 Expansion of Product_{k >= 1} (1-x^k)^48.

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cp's OEIS Frontend

A290010 Coefficients in expansion of 691*E_6^2*E_12.

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Author

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Mathematica

A010839 Expansion of Product_{k >= 1} (1-x^k)^48.

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Author

Keywords

Examples

References

Links

Crossrefs

Formula

A290010 Coefficients in expansion of 691E_6^2E_12.