A172493 E.g.f. satisfies: A(x) = Sum_{n>=0} AGM(1, A(x)^(4n))*x^n/n!, where AGM(x,y) is the arithmetic-geometric mean of Gauss.

1, 1, 5, 55, 969, 23471, 722893, 27025349, 1188914961, 60185489239, 3446702343621, 220325043859361, 15551414491260409, 1201309497935878085, 100806806760636877293, 9131452009580323562311, 888090470343071154122145

Offset

0,3

Comments

a(61) is negative. - Vaclav Kotesovec, Mar 31 2024

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! +...
The e.g.f. satisfies the series:
A(x) = 1 + AGM(1,A(x)^4)*x + AGM(1,A(x)^8)*x^2/2! + AGM(1,A(x)^12)*x^3/3! + AGM(1,A(x)^16)*x^4/4! +...
In series expansions of AGM(1,A(x)^(4n)), the coefficients of x^k/k! for n=1..8 begin:
n=1: [1, 2, 14, 176, 3298, 82872, 2618340, 99766088, ...];
n=2: [1, 4, 40, 616, 12992, 352104, 11734032, ...];
n=3: [1, 6, 78, 1440, 34338, 1013736, 36005076, ...];
n=4: [1, 8, 128, 2768, 74176, 2388048, 90792672, ...];
n=5: [1, 10, 190, 4720, 140930, 4935000, 201048420, ...];
n=6: [1, 12, 264, 7416, 244608, 9279672, 404745840, ...];
n=7: [1, 14, 350, 10976, 396802, 16237704, 756856212, ..];
n=8: [1, 16, 448, 15520, 610688, 26840736, 1333868736, ...].

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(k=1, n, agm(1, (A+x*O(x^n))^(4*k))*x^k/k!)); n!*polcoeff(A, n)}

Crossrefs

Sequence in context: A093352 A293013 A195513 * A155807 A135861 A141361

Adjacent sequences: A172490 A172491 A172492 * A172494 A172495 A172496

Keyword

sign

Author

Paul D. Hanna, Jan 26 2011

Status

approved