A341380 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x^(2*n+1)/A(x)).

1, 2, 2, 6, 19, 58, 198, 680, 2410, 8695, 31870, 118377, 444315, 1683400, 6428086, 24715541, 95603500, 371784813, 1452687192, 5700329627, 22454015652, 88755923251, 351944894971, 1399612973849, 5580765692117, 22306991852511, 89365286885821

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

G.f. A(x) satisfies:

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

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

(3) A(x) = Sum_{n>=0} x^(2*n*(n+1)) * (1 - x^(4*n+2)) / ((1 - x^(2*n+1)*A(x))*(1 - x^(2*n+1)/A(x)).

a(n) ~ c * d^n / n^(3/2), where d = 4.24149290765489... and c = 0.598891272666... - Vaclav Kotesovec, Mar 03 2021

Example

G.f.: A(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 19*x^4 + 58*x^5 + 198*x^6 + 680*x^7 + 2410*x^8 + 8695*x^9 + 31870*x^10 + 118377*x^11 + 444315*x^12 + ...
where
A(x) = 1/(1 - x/A(x)) + x*A(x)/(1 - x^3/A(x)) + x^2*A(x)^2/(1 - x^5/A(x)) + x^3*A(x)^3/(1 - x^7/A(x)) + x^4*A(x)^4/(1 - x^9/A(x)) + ...
also
A(x) = 1/(1 - x*A(x)) + (x/A(x))/(1 - x^3*A(x)) + (x/A(x))^2/(1 - x^5*A(x)) + (x/A(x))^3/(1 - x^7*A(x)) + (x/A(x))^4/(1 - x^9*A(x)) + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, n, x^m*A^m/(1 - x^(2*m+1)/A +x*O(x^n)) ); );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, n, x^m/A^m/(1 - x^(2*m+1)*A +x*O(x^n)) ); );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, sqrtint(n), x^(2*m*(m+1)) * (1 - x^(4*m+2)) / ((1 - x^(2*m+1)*A)*(1 - x^(2*m+1)/A) +x*O(x^n)) ); );
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A370836 A233113 A366731 * A171694 A103179 A320932

Adjacent sequences: A341377 A341378 A341379 * A341381 A341382 A341383

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2021

Status

approved