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%I #17 Aug 16 2014 06:01:23
%S 1,2,3,4,9,30,91,232,549,1378,3839,11100,31301,85694,234207,653328,
%T 1856829,5300010,15062839,42702596,121448901,347414166,997886671,
%U 2870139480,8257776521,23782773242,68627659563,198437633884,574654851209,1665825647430,4833258038251,14037680955552,40816416373293
%N G.f.: 1 / AGM((1 - 3*x)^2, (1 + x)^2).
%C Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%C Self-convolution of A245931.
%C Limit a(n+1)/a(n) = 3.
%H Paul D. Hanna, <a href="/A245930/b245930.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1 / AGM((1-x)^2 + 4*x^2, (1-x)^2 - 4*x^2).
%F G.f.: 1 / AGM((1-x)^2, sqrt((1-x)^4 - 16*x^4)).
%F Recurrence: n^2*a(n) = (5*n^2 - 5*n + 2)*a(n-1) - 2*(5*n^2 - 10*n + 6)*a(n-2) + 2*(5*n^2 - 15*n + 12)*a(n-3) + 11*(n-2)^2*a(n-4) - 15*(n-3)*(n-2)*a(n-5). - _Vaclav Kotesovec_, Aug 16 2014
%F a(n) ~ 3^(n+2) / (4*Pi*n). - _Vaclav Kotesovec_, Aug 16 2014
%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 9*x^4 + 30*x^5 + 91*x^6 + 232*x^7 +...
%e where
%e 1/A(x) = 1 - 2*x + x^2 - 4*x^4 - 8*x^5 - 12*x^6 - 16*x^7 - 40*x^8 - 144*x^9 - 448*x^10 - 1152*x^11 - 2732*x^12 - 6840*x^13 - 18964*x^14 +...
%e equals AGM((1 - 3*x)^2, (1 + x)^2).
%e SPECIFIC VALUES:
%e A(x) = 2 at x = 0.2650276124990406644...
%e A(x) = 3 at x = 0.31872724866867463...
%e A(x) = 4 at x = 0.32990867978741...
%o (PARI) {a(n)=local(A=1); A = 1 / agm((1-3*x)^2, (1+x)^2 +x*O(x^n)); polcoeff(A,n)}
%o for(n=0,35,print1(a(n),", "))
%Y Cf. A245931, A245932, A227845.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 14 2014
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