OFFSET
0,3
COMMENTS
a(n) = Sum_{k=0..n} A355350(n,k) for n >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..600
FORMULA
G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/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) = QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)).
a(n) ~ c * d^n / n^(3/2), where d = 5.163920888936085556632234304058129... and c = 0.824708825453794494929019119272... (End)
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + 16365*x^8 + 71792*x^9 + 320219*x^10 + 1446517*x^11 + ...
where
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 -+ ...
also,
x*P(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)) * ...
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, s*Sqrt[(-1 + s)*(-1 + r*s) * Log[r]* ((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r] - r*(-1 + s)*(-1 + r*s)*Log[r]* Derivative[0, 1][QPochhammer][r, r] / QPochhammer[r] + r*s*Log[r] * QPochhammer[r] * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + (-1 + r*s) * ((1 - s) * QPolyGamma[0, Log[s]/Log[r], r] + Log[r] * (-1 - (r*(-1 + s) * Derivative[0, 1][QPochhammer][s, 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, -Log[r*s]/Log[r], r]) / (2*Pi)]} /. FindRoot[{1/QPochhammer[r] + s*QPochhammer[1/(r*s), r] * QPochhammer[s, r] / ((-1 + s) * (-1 + r*s)) == 0, (-1 + r*s^2)*Log[r] + (-1 + s) * (-1 + r*s) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s] / Log[r], r]) == 0}, {r, 1/5}, {s, 2}, 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(2*n+9));
A[#A] = -polcoeff( 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, Jun 29 2022
STATUS
approved