A360191 - OEIS

%I #12 Mar 19 2023 14:40:41

%S 1,5,18,55,149,371,867,1923,4086,8374,16634,32152,60669,112041,202943,

%T 361200,632647,1091917,1859225,3126242,5195715,8541624,13899866,

%U 22404091,35787815,56683294,89061028,138872410,214984454,330532633,504869316,766357010,1156355165

%N G.f. 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2.

%C Self-convolution inverse of A080332.

%H Vaclav Kotesovec, <a href="/A360191/b360191.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

%F (!) A(x) = 1 / [Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2].

%F (2) A(x) = 1 / [Sum_{n=-oo..+oo} (6*n + 1) * x^(n*(3*n + 1)/2)].

%F a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (12*sqrt(2)*n^(3/2)). - _Vaclav Kotesovec_, Feb 07 2023

%e G.f.: A(x) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + 16634*x^10 + 32152*x^11 + 60669*x^12 + ...

%t nmax = 30; CoefficientList[Series[1/Product[(1 - x^k)^3 * (1 - x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 07 2023 *)

%t nmax = 30; CoefficientList[Series[1/(QPochhammer[x] * EllipticTheta[4, 0, x]^2), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 07 2023 *)

%o (PARI) {a(n) = polcoeff( 1/prod(m=1,n, (1 - x^m)^3 * (1 - x^(2*m-1))^2 +x*O(x^n)), n)}

%o for(n=0,32,print1(a(n),", "))

%Y Cf. A080332, A361535, A361050, A361550.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 29 2023