A352120 G.f. A(x) satisfies: Product_{n>=1} (1 + x^n*A(x)) = Product_{n>=1} (1 + x^n/(1-x)^n).

1, 1, 2, 5, 10, 20, 43, 93, 194, 403, 842, 1755, 3656, 7643, 15976, 33281, 69164, 143558, 297619, 616625, 1277729, 2647861, 5485300, 11356731, 23495794, 48567063, 100301668, 206994479, 426941231, 880227976, 1814221503, 3738368348, 7701376466

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n and P(x) = Product_{n>=1} (1 + x^n/(1-x)^n) satisfies:

(1) P(x) = Product_{n>=1} (1 + x^n*A(x)).

(2) P(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n / (Product_{k=1..n} (1 - x^k)).

(3) 1/P(x) = Sum_{n>=0} (-x)^n * A(x)^n / (Product_{k=1..n} (1 - x^k)).

(4) log(P(x)) = Sum_{n>=1} x^n * Sum_{d|n} -(-A(x))^(n/d) * d/n.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 20*x^5 + 43*x^6 + 93*x^7 + 194*x^8 + 403*x^9 + 842*x^10 + 1755*x^11 + 3656*x^12 + ...
such that the following products are equal:
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)) * (1 + x^6*A(x)) * ...
P(x) = (1 + x/(1-x)) * (1 + x^2/(1-x)^2) * (1 + x^3/(1-x)^3) * (1 + x^4/(1-x)^4) * (1 + x^5/(1-x)^5) * ...
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)) + ...
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)) -+ ...
where
P(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 28*x^5 + 65*x^6 + 151*x^7 + 350*x^8 + 807*x^9 + 1850*x^10 + ... + A129519(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( prod(n=1, #A, (1 + x^n/(1-x +x*O(x^#A))^n)/(1 + x^n*Ser(A)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A129519.

Sequence in context: A006836 A129847 A330456 * A176692 A212951 A051109

Adjacent sequences: A352117 A352118 A352119 * A352121 A352122 A352123

Keyword

nonn

Author

Paul D. Hanna, Mar 05 2022

Status

approved