OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..520
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1.a) P(x) = Product_{n>=1} 1 + x^n*A(x).
(1.b) P(x)*(1-x) = Product_{n>=1} 1 + x^(2*n)*A(x)^2.
(2.a) P(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n / (Product_{k=1..n} 1 - x^k).
(2.b) P(x)*(1-x) = Sum_{n>=0} x^(n*(n+1)) * A(x)^(2*n) / (Product_{k=1..n} 1 - x^(2*k)).
(3.a) 1/P(x) = Sum_{n>=0} (-x)^n * A(x)^n / (Product_{k=1..n} 1 - x^k).
(3.b) 1/(P(x)*(1-x)) = Sum_{n>=0} (-x^2)^n * A(x)^(2*n) / (Product_{k=1..n} 1 - x^(2*k)).
(4.a) log(P(x)) = Sum_{n>=1} x^n * Sum_{d|n} -(-A(x))^(n/d) * d/n.
(4.b) log(P(x)*(1-x)) = Sum_{n>=1} x^(2*n) * Sum_{d|n} -(-A(x)^2)^(n/d) * d/n.
See the example section for the power series expansion of P(x).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 22*x^6 + 57*x^7 + 150*x^8 + 401*x^9 + 1086*x^10 + 2972*x^11 + 8211*x^12 + ...
such that the following products hold:
P(x) = (1 + x*A(x)) * (1 + x^2*A(x)) * (1 + x^3*A(x)) * (1 + x^4*A(x)) * (1 + x^5*A(x)) * ...
P(x)*(1-x) = (1 + x^2*A(x)^2) * (1 + x^4*A(x)^2) * (1 + x^6*A(x)^2) * (1 + x^8*A(x)^2) * (1 + x^10*A(x)^2) * ...
also, we have the sums
P(x) = 1 + x*A(x)/(1-x) + x^3*A(x)^2/((1-x)*(1-x^2)) + x^6*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...
P(x)*(1-x) = 1 + x^2*A(x)^2/(1-x^2) + x^6*A(x)^4/((1-x^2)*(1-x^4)) + x^12*A(x)^6/((1-x^2)*(1-x^4)*(1-x^6)) + x^20*A(x)^8/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)) + ...
1/P(x) = 1 - x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-x^2)) - x^3*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^4*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) -+ ...
1/(P(x)*(1-x)) = 1 - x^2*A(x)^2/(1-x^2) + x^4*A(x)^4/((1-x^2)*(1-x^4)) - x^6*A(x)^6/((1-x^2)*(1-x^4)*(1-x^6)) + x^8*A(x)^8/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)) -+ ...
where
P(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 34*x^6 + 76*x^7 + 178*x^8 + 436*x^9 + 1108*x^10 + 2894*x^11 + 7718*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( prod(n=1, #A, (1 + x^(2*n)*Ser(A)^2)/(1 + x^n*Ser(A)) ), #A) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 07 2022
STATUS
approved