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A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8. 8

%I #11 Feb 20 2021 06:35:42

%S 1,6,24,77,216,552,1315,2964,6387,13255,26640,52074,99336,185430,

%T 339483,610709,1081227,1886484,3247502,5521365,9279624,15429149,

%U 25397088,41412030,66928700,107265576,170556654,269164346,421765920,656419080,1015044526,1559950185,2383284894

%N Expansion of (1 / theta_4(x) - 1)^3 / 8.

%H Alois P. Heinz, <a href="/A341364/b341364.txt">Table of n, a(n) for n = 3..10000</a>

%F G.f.: (1/8) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^3.

%F a(n) ~ A319552(n)/8 ~ 3*exp(Pi*sqrt(3*n)) / (512*n^(3/2)). - _Vaclav Kotesovec_, Feb 20 2021

%p g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,

%p g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))

%p end:

%p b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,

%p g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))

%p end:

%p a:= n-> b(n, 3):

%p seq(a(n), n=3..35); # _Alois P. Heinz_, Feb 10 2021

%t nmax = 35; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^3/8, {x, 0, nmax}], x] // Drop[#, 3] &

%t nmax = 35; CoefficientList[Series[(1/8) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

%Y Cf. A002448, A004404, A014968, A015128, A063691, A319552, A327381, A338223, A341221, A341365, A341366, A341367, A341368, A341369, A341370.

%K nonn

%O 3,2

%A _Ilya Gutkovskiy_, Feb 10 2021

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)