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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

%I #36 Mar 31 2024 08:47:55

%S 1,1,5,55,969,23471,722893,27025349,1188914961,60185489239,

%T 3446702343621,220325043859361,15551414491260409,1201309497935878085,

%U 100806806760636877293,9131452009580323562311,888090470343071154122145

%N 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.

%C a(61) is negative. - _Vaclav Kotesovec_, Mar 31 2024

%H Vaclav Kotesovec, <a href="/A172493/b172493.txt">Table of n, a(n) for n = 0..100</a>

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! +...

%e The e.g.f. satisfies the series:

%e 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! +...

%e In series expansions of AGM(1,A(x)^(4n)), the coefficients of x^k/k! for n=1..8 begin:

%e n=1: [1, 2, 14, 176, 3298, 82872, 2618340, 99766088, ...];

%e n=2: [1, 4, 40, 616, 12992, 352104, 11734032, ...];

%e n=3: [1, 6, 78, 1440, 34338, 1013736, 36005076, ...];

%e n=4: [1, 8, 128, 2768, 74176, 2388048, 90792672, ...];

%e n=5: [1, 10, 190, 4720, 140930, 4935000, 201048420, ...];

%e n=6: [1, 12, 264, 7416, 244608, 9279672, 404745840, ...];

%e n=7: [1, 14, 350, 10976, 396802, 16237704, 756856212, ..];

%e n=8: [1, 16, 448, 15520, 610688, 26840736, 1333868736, ...].

%o (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)}

%K sign

%O 0,3

%A _Paul D. Hanna_, Jan 26 2011

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