A341370 - OEIS

%I #9 Feb 10 2021 11:29:22

%S 1,18,180,1311,7740,39204,176388,721530,2728053,9651056,32246892,

%T 102515508,311923386,912771468,2579132196,7060677537,18781247700,

%U 48660380190,123061973176,304351869708,737293187286,1752035386188,4089222211212,9384936015492,21201250825554

%N Expansion of (1 / theta_4(x) - 1)^9 / 512.

%H Alois P. Heinz, <a href="/A341370/b341370.txt">Table of n, a(n) for n = 9..10000</a>

%F G.f.: (1/512) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^9.

%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, 9):

%p seq(a(n), n=9..33);  # _Alois P. Heinz_, Feb 10 2021

%t nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^9/512, {x, 0, nmax}], x] // Drop[#, 9] &

%t nmax = 33; CoefficientList[Series[(1/512) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

%Y Cf. A002448, A004410, A014968, A015128, A327387, A338223, A340946, A341228, A341364, A341365, A341366, A341367, A341368, A341369.

%K nonn

%O 9,2

%A _Ilya Gutkovskiy_, Feb 10 2021