OFFSET
0,2
COMMENTS
a(n) = Sum_{k=0..n} A355350(n,k) * 3^k for n >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) satisfies:
(1) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 3*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.
EXAMPLE
G.f.: A(x) = 1 + 3*x + 18*x^2 + 108*x^3 + 660*x^4 + 4275*x^5 + 29106*x^6 + 205377*x^7 + 1485279*x^8 + 10943424*x^9 + 81866493*x^10 + ...
where
3*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,
3*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 + ...
PROG
(PARI) {a(n) = my(A=[1, 3], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*(#A)+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