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Absolute values of coefficients of an elliptic function.
(Formerly M4411 N1864)
6

%I M4411 N1864 #26 Dec 04 2017 09:17:21

%S 1,7,35,140,483,1498,4277,11425,28889,69734,161735,362271,786877,

%T 1662927,3428770,6913760,13660346,26492361,50504755,94766875,

%U 175221109,319564227,575387295,1023624280,1800577849,3133695747,5399228149,9214458260,15584195428

%N Absolute values of coefficients of an elliptic function.

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001941/b001941.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, 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]

%F G.f.: Product ( 1 - x^k )^-c(k), c(k) = 7, 7, 7, 0, 7, 7, 7, 0, ....

%F a(n) ~ 7^(1/4) * exp(sqrt(7*n/2)*Pi) / (256*2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017

%F G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^7. - _Ilya Gutkovskiy_, Dec 04 2017

%t nn = 4*10; b = Flatten[Table[{7, 7, 7, 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 *)

%t nmax = 40; CoefficientList[Series[Product[((1 - x^(4*k)) / (1 - x^k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 15 2017 *)

%Y Cf. A001935, A001936, A001937, A001939, A001940, A092877, A093160.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_