A001940 Absolute value of coefficients of an elliptic function.

1, 6, 27, 98, 309, 882, 2330, 5784, 13644, 30826, 67107, 141444, 289746, 578646, 1129527, 2159774, 4052721, 7474806, 13569463, 24274716, 42838245, 74644794, 128533884, 218881098, 368859591, 615513678, 1017596115, 1667593666, 2710062756, 4369417452

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

References

• A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]

Crossrefs

Cf. A001935, A001936, A001937, A001939, A001941, A092877, A093160.

Programs

Mathematica

nn = 4*10; b = Flatten[Table[{6, 6, 6, 0}, {nn/4}]]; CoefficientList[x*Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
nmax = 40; CoefficientList[Series[Product[((1 - x^(4*k)) / (1 - x^k))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2017 *)

Formula

G.f.: Product ( 1 - x^k )^(-c(k)), c(k) = 6, 6, 6, 0, 6, 6, 6, 0, ....

a(n) ~ 3^(1/4) * exp(sqrt(3*n)*Pi) / (128*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^6. - Ilya Gutkovskiy, Dec 04 2017

Extensions

Extended and corrected by Simon Plouffe