A355862 G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * (x^n - 2*A(x))^(n+1).

1, 2, 6, 25, 112, 557, 2914, 15837, 88531, 505581, 2936676, 17294352, 103018292, 619595991, 3757342674, 22948207189, 141033508661, 871527612640, 5412015056754, 33754524947592, 211353845133650, 1328099943458743, 8372466442163468, 52936608451071755

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 - 2*A(x))^(n+1).

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

a(n) ~ c * d^n / n^(3/2), where d = 6.74709799536602052858389740164829219437... and c = 0.517304287814827280375970612560243586... - Vaclav Kotesovec, Jul 23 2022

A(1/d) = 2.022729610323037319... where 1/d = 0.148211868374642... and d is the value given above by Vaclav Kotesovec. - Paul D. Hanna, Jul 30 2022

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 25*x^3 + 112*x^4 + 557*x^5 + 2914*x^6 + 15837*x^7 + 88531*x^8 + 505581*x^9 + 2936676*x^10 + 17294352*x^11 + ...
where
0 = ... + x^6/(1/x^4 - 2*A(x))^3+ x^3/(1/x^3 - 2*A(x))^2 + x/(1/x^2 - 2*A(x)) + 1 + (1 - 2*A(x)) + x*(x - 2*A(x))^2 + x^3*(x^2 - 2*A(x))^3 + x^6*(x^3 - A(x))^4 + ... + x^(n*(n+1)/2)*(x^n - 2*A(x))^(n+1) + ...
Specific values.
A(0.148188601...) = 2.
A(1/7) = 1.72240285856328...

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 - 2*Ser(A))^(m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A199241 A366236 A276277 * A299098 A229042 A269484

Adjacent sequences: A355859 A355860 A355861 * A355863 A355864 A355865

Keyword

nonn

Author

Paul D. Hanna, Jul 22 2022

Status

approved