A352176 G.f. A(X) satisfies: 1 - x = Product_{n>=1} (1 + x^(2*n)*A(x)^2) / (1 + x^n*A(x)).

1, 1, 1, 2, 4, 9, 22, 57, 150, 401, 1086, 2972, 8211, 22882, 64250, 181608, 516347, 1475714, 4237134, 12216498, 35355041, 102668187, 299068056, 873654587, 2558849874, 7512707796, 22106301618, 65182852893, 192568585013, 569922496063, 1689559407054

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

Sequence in context: A307575 A301362 A130018 * A322504 A099754 A105633

Adjacent sequences: A352173 A352174 A352175 * A352177 A352178 A352179

Keyword

nonn

Author

Paul D. Hanna, Mar 07 2022

Status

approved