OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) satisfies:
(1) x/P(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n, where P(x) = 1/Product_{n>=1} (1 - x^n) is the partition function (A000041).
(2) x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 4.6003483603736784205277234... and c = 0.69610758028428020320488... - Vaclav Kotesovec, Oct 04 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 32*x^4 + 108*x^5 + 382*x^6 + 1419*x^7 + 5437*x^8 + 21288*x^9 + 84618*x^10 + 340499*x^11 + 1384711*x^12 + ...
such that
x = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * (1 - x^5*A(x))*(1 - x^4/A(x)) * ...
also,
x/P(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
and
x/P(x) = x - x^2 - x^3 + x^6 + x^8 - x^13 - x^16 + x^23 + x^27 - x^36 - x^41 + x^52 + x^58 - x^71 - x^78 + x^93 + x^101 + ...
The following expressions involving g.f. A(x) are all equal:
B(x) = 1 / Product_{n>=1} (1 - x^n*A(x)),
B(x) = (1/x) * Product_{n>=1} (1 - x^(n-1)/A(x)),
B(x) = Sum_{n>=0} x^n * A(x)^n / Product_{k=1..n} (1 - x^k),
B(x) = ((1 - 1/A(x))/x) / [Sum_{n>=0} (x/A(x))^n/Product_{k=1..n} (1 - x^k)],
where
B(x) = 1 + x + 3*x^2 + 9*x^3 + 31*x^4 + 109*x^5 + 396*x^6 + 1472*x^7 + 5613*x^8 + 21868*x^9 + 86690*x^10 + 348422*x^11 + 1416090*x^12 + 5809655*x^13 + 24028116*x^14 + 100081147*x^15 + ...
MATHEMATICA
(* Calculation of constants {d, c}: *) {1/r, Sqrt[(-r)*(((-1 + s)*s^2*(-1 + r*s)* Log[r]*((-1 + r*s) * QPochhammer[s, r]*((-1 + s)*Log[1 - r] + Log[r] + (-1 + s)*QPolyGamma[0, Log[s]/Log[r], r]) - r*s*Log[r]*QPochhammer[s, r]^2*Derivative[0, 1][QPochhammer][1/(r*s), r] + r*(-1 + s)*(-1 + r*s)*Log[r] * Derivative[0, 1][QPochhammer][s, r])) / (2* Pi*(r*QPochhammer[s, r]*(-s*(1 + r - 4*r*s + r*(1 + r)*s^2)* Log[r]^2 + (-1 + s)^2*(-1 + r*s)^2* QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)^2*(-1 + r*s)^2 * QPolyGamma[1, -1 - Log[s]/Log[r], r]))))]} /. FindRoot[{s*QPochhammer[1/(r*s), r]* QPochhammer[s, r] == (s - 1)*(1 - r*s), (-1 + s)*(-1 + r*s)*(QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -1 - Log[s]/Log[r], r]) == (1 - r*s^2)*Log[r]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x - prod(n=1, #A, (1 - x^n*Ser(A)) * (1 - x^(n-1)/Ser(A)) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1], t, P=prod(k=1, n, 1-x^k +x*O(x^n))); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*P - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 11 2022
STATUS
approved