OFFSET
0,4
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) x*P(x^2) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
From Vaclav Kotesovec, Feb 01 2024: (Start)
Formula (2) can be rewritten as the functional equation x/QPochhammer(x^2) = QPochhammer(y, x^2)/(1 - y) * QPochhammer(1/(x^2*y), x^2)/(1 - 1/(x^2*y)).
a(n) ~ c * d^n / n^(3/2), where d = 2.92005174190265697439941308343193651904071627244119127019370275824199... and c = 1.4709989760845501303394202030872391136773745007487301056274536584990... (End)
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 20*x^5 + 43*x^6 + 110*x^7 + 262*x^8 + 674*x^9 + 1684*x^10 + 4397*x^11 + 11320*x^12 + ...
where
x = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 -+ ...
also,
x*P(x^2) = (1 - x^2*A(x))*(1 - 1/A(x)) * (1 - x^4*A(x))*(1 - x^2/A(x)) * (1 - x^6*A(x))*(1 - x^4/A(x)) * (1 - x^8*A(x))*(1 - x^6/A(x)) * ...
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 + ...
MATHEMATICA
(* Calculation of constants {d, c}: *) {1/r, Sqrt[-Log[r] * ((-1 + r) * QPochhammer[1/r, r^2] * (-2*Log[r] + (-1 + r)*(Log[1 - r^2] - Log[r - r^3]) + (-1 + r) * QPolyGamma[0, -1/2, r^2] - (-1 + r)*QPolyGamma[0, 1, r^2]) + 4*(-1 + r)^2 * r^2 * Log[r] * Derivative[0, 1][QPochhammer][1/r, r^2] + 2*r^3 * Log[r] * QPochhammer[1/r, r^2]^3 * Derivative[0, 1][QPochhammer][r^2, r^2]) / (Pi*r^2* QPochhammer[1/r, r^2] * (-4*r*Log[r]^2 + (-1 + r)^2 * QPolyGamma[1, -1/2, r^2]))]} /. FindRoot[ 1/QPochhammer[r^2] == (r*QPochhammer[1/r, r^2]^2)/(-1 + r)^2, {r, 1/3}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 01 2024 *)
PROG
(PARI) {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(n+4));
A[#A] = -polcoeff( sum(m=-t, t, (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 29 2022
STATUS
approved