A001940 Absolute value of coefficients of an elliptic function. (Formerly M4173 N1737)

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

0,2

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

Arthur 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]

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 15/24, marked 12.

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * exp(3*Pi/4) = A388099. - Simon Plouffe, Sep 14 2025

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 *)

Crossrefs

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

Sequence in context: A344100 A136747 A278357 * A320049 A121591 A071734

Adjacent sequences: A001937 A001938 A001939 * A001941 A001942 A001943

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Extended and corrected by Simon Plouffe

Status

approved