A359462 a(n) = coefficient of x^n/n! in A(x) = Sum_{n>=0} x^n * ( (A(x)^sqrt(2*n) + x)^sqrt(2*n) + A(x)^(2*n)/(1 + x*A(x)^sqrt(2*n))^sqrt(2*n) )/2.

1, 1, 6, 66, 1080, 24210, 689160, 23806160, 967458688, 45226555164, 2391009969600, 141058577704968, 9187128380964864, 654761426765669080, 50687036874968329472, 4235395789572088071840, 379953548646431877841920, 36422572996966918108870800, 3715607345159938499844931584

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.

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

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

Example

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1080*x^4/4! + 24210*x^5/5! + 689160*x^6/6! + 23806160*x^7/7! + 967458688*x^8/8! + 45226555164*x^9/9! + 2391009969600*x^10/10! + ...
SPECIFIC VALUES.
A(x) = sqrt(2) at x = 0.156934823890276913061148911866717130169736795253864482...
A(2/13) = 1.37929578566666389269739402377879264233420917786313229616...
A(1/7) = 1.29917227096629410743551160817342310342620062625852904026...
A(1/8) = 1.22070880501883803045142406156799735607641427440734581159...

Prog

(PARI) /* must set precision suitable for desired number of terms */
\p60
{a(n) = my(A=1+x); for(i=1, 30, A = sum(m=0, n, x^m/2 * ( ((A +x*O(x^n))^sqrt(2*m) + x)^sqrt(2*m) + A^(2*m)/(1 + x*(A +x*O(x^n))^sqrt(2*m) )^sqrt(2*m) ) )); round(n!*polcoeff(H=A, n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A359461.

Sequence in context: A174496 A008548 A090358 * A264407 A112942 A113390

Adjacent sequences: A359459 A359460 A359461 * A359463 A359464 A359465

Keyword

nonn

Author

Paul D. Hanna, Jan 03 2023

Status

approved