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Expansion of (1 / theta_4(x) - 1)^6 / 64.
7

%I #7 Feb 10 2021 10:04:07

%S 1,12,84,442,1932,7392,25551,81468,243126,686400,1848156,4775874,

%T 11904215,28737732,67416756,154122912,344177823,752310720,1612395007,

%U 3393652848,7023685794,14311193104,28737793986,56924936052,111323290934,215095157964,410895944148,776529566516

%N Expansion of (1 / theta_4(x) - 1)^6 / 64.

%F G.f.: (1/64) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^6.

%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, 6):

%p seq(a(n), n=6..33); # _Alois P. Heinz_, Feb 10 2021

%t nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &

%t nmax = 33; CoefficientList[Series[(1/64) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

%Y Cf. A002448, A004407, A014968, A015128, A327384, A338223, A340905, A341225, A341364, A341365, A341366, A341368, A341369, A341370.

%K nonn

%O 6,2

%A _Ilya Gutkovskiy_, Feb 10 2021