A355861 G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} x^(n*(n-1)/2) * (x^n - A(x))^n.

1, 2, 5, 12, 37, 121, 419, 1510, 5604, 21261, 82110, 321662, 1275077, 5104886, 20611814, 83834609, 343164051, 1412600336, 5843868040, 24283650452, 101312783192, 424212909937, 1782086178267, 7508852850710, 31725558330499, 134381573170076, 570532128884181

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) satisfies:

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

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

a(n) ~ c * d^n / n^(3/2), where d = 4.4984524719254844098592086179118472526... and c = 0.813659600743159302516036718242855... - Vaclav Kotesovec, Jul 23 2022

A(1/d) = 2.6481605623378565... where 1/d = 0.2222986696516029605295343... and d is the value given above by Vaclav Kotesovec. - Paul D. Hanna, Jul 28 2022

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 37*x^4 + 121*x^5 + 419*x^6 + 1510*x^7 + 5604*x^8 + 21261*x^9 + 82110*x^10 + 321662*x^11 + ...
where
0 = ... + x^6/(1/x^3 - A(x))^3 + x^3/(1/x^2 - A(x))^2 + x/(1/x - A(x)) + 1 + (x - A(x)) + x*(x^2 - A(x))^2 + x^3*(x^3 - A(x))^3 + x^6*(x^4 - A(x))^4 + ... + x^(n*(n-1)/2)*(x^n - A(x))^n + ...
Specific values.
A(0.20604788...) = 2.
A(1/5) = 1.9097493689592466...

Prog

(PARI) {a(n) = my(A=[1], M); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( sum(m=-M, M, x^(m*(m-1)/2) * (x^m - Ser(A))^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A052302 A280275 A009598 * A002216 A024717 A003724

Adjacent sequences: A355858 A355859 A355860 * A355862 A355863 A355864

Keyword

nonn

Author

Paul D. Hanna, Jul 22 2022

Status

approved