login
G.f. A(x) satisfies: 1 = Product_{n>=1} (1 - x^n) * (1 - x^(n+1)/A(x)) * (1 - x^(n-2)*A(x)).
2

%I #14 Sep 30 2023 04:58:33

%S 1,1,2,5,9,22,52,146,377,1036,2810,8014,22790,66100,191541,562926,

%T 1660975,4944766,14767136,44357952,133698623,404810569,1229434572,

%U 3746595869,11447723074,35075829156,107724187826,331605018200,1022842337041,3161156987190,9787096605716,30352665554591,94279407445012,293277650593792,913565090912339,2849489942324466,8898714901181309,27822251614174021,87083081436755770

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

%C The g.f. utilizes the Jacobi Triple Product:

%C Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

%H Paul D. Hanna, <a href="/A268651/b268651.txt">Table of n, a(n) for n = 0..512</a>

%F G.f. A(x) satisfies:

%F (1) x = Sum_{n=-oo..oo} (-1)^n * x^((n-1)*(n-2)/2) * A(x)^n.

%F (2) x = sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (A(x)^(2*n+1) - 1) / A(x)^(n-1).

%F (3) A(x) = x / Series_Reversion( G(x) ), where G(x) is the g.f. of A268650.

%F a(n) ~ c * d^n / n^(3/2), where d = 3.25766000970998791... and c = 0.661369655158037... . - _Vaclav Kotesovec_, Mar 05 2016

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 22*x^5 + 52*x^6 + 146*x^7 + 377*x^8 + 1036*x^9 + 2810*x^10 + 8014*x^11 + 22790*x^12 + 66100*x^13 + 191541*x^14 + 562926*x^15 +...

%e where A(x) satisfies the Jacobi Triple Product:

%e 1 = (1-x)*(1-x^2/A(x))*(1-1/x*A(x)) * (1-x^2)*(1-x^3/A(x))*(1-1*A(x)) * (1-x^3)*(1-x^4/A(x))*(1-x*A(x)) * (1-x^4)*(1-x^5/A(x))*(1-x^2*A(x)) * (1-x^5)*(1-x^6/A(x))*(1-x^3*A(x)) * (1-x^6)*(1-x^7/A(x))*(1-x^4*A(x)) *...

%e Also

%e x = (A(x)-1)*A(x) - x*(A(x)^3-1) + x^3*(A(x)^5-1)/A(x) - x^6*(A(x)^7-1)/A(x)^2 + x^10*(A(x)^9-1)/A(x)^3 - x^15*(A(x)^11-1)/A(x)^4 + x^21*(A(x)^13-1)/A(x)^5 +...

%t (* Calculation of constant d: *) 1/r /. FindRoot[{s*QPochhammer[r, r] * QPochhammer[r/s, r] * QPochhammer[s/r^2, r] == (s - r)*(1 - s/r^2), (r^3 - s^2)* Log[r] + (r^3 - r*s - r^2*s + s^2) * (QPolyGamma[0, Log[r/s]/Log[r], r] - QPolyGamma[0, Log[s/r^2]/Log[r], r]) == 0}, {r, 1/3}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Sep 30 2023 *)

%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); A[#A]=-Vec( sum(m=1,sqrtint(2*#A)+2,(-1)^m*(x*Ser(A))^(m*(m-1)/2)*(1-x^(2*m-1))/x^m) )[#A-1] );Vec(x/serreverse(x*Ser(A)))[n+1]}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A268650.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 02 2016