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%I #16 May 31 2014 20:36:55
%S 1,1,3,17,151,1901,31841,679243,17873349,566127595,21172659297,
%T 920475938637,45922819496273,2600893043805459,165687190863751905,
%U 11778064492639412479,927828923151187295125,80505341430961590290171,7652425565871036374257731,793045506373759916564428397
%N Coefficients of power series generated by the continued fraction-like form: A(x) = 1/(1 - x/sqrt(1 - 4x/cube-root(1 - 9x/4th-root(1 - 16x/5th-root(1 - 25x/... /n-th-root(1 - (n^2)x/...)))))), where n-th-root(z)=z^(1/n).
%H Paul D. Hanna, <a href="/A075820/b075820.txt">Table of n, a(n) for n = 0..200</a>
%F A(x) = F(0,x) where F(0,x) = 1/(1 - x*F(1,x)), F(1,x) = 1/(1 - 4*x*F(2,x))^(1/2), F(2,x) = 1/(1 - 9*x/F(3,x))^(1/3), F(3,x) = 1/(1 - 16*x/F(4,x))^(1/4), ..., so that F(n,x) = 1/(1 - (n+1)^2*x*F(n+1,x))^(1/(n+1)) for n>=0.
%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1901*x^5 + 31841*x^6 +...
%e Define F(n,x) with F(0,x) = A(x) where for n>0 we have
%e F(n,x) = (1 - 1/F(n-1,x)^n)/(n^2*x) = 1/(1 - (n+1)^2*x*F(n+1,x))^(1/(n+1)),
%e then
%e F(1,x) = 1 + 2*x + 12*x^2 + 116*x^3 + 1564*x^4 + 27522*x^5 +...
%e F(2,x) = 1 + 3*x + 30*x^2 + 450*x^3 + 8805*x^4 + 212559*x^5 +...
%e F(3,x) = 1 + 4*x + 60*x^2 + 1300*x^3 + 35300*x^4 + 1139704*x^5 +...
%e F(4,x) = 1 + 5*x + 105*x^2 + 3115*x^3 + 112910*x^4 + 4747680*x^5 +...
%e F(5,x) = 1 + 6*x + 168*x^2 + 6552*x^3 + 306936*x^4 + 16398900*x^5 +...
%e F(6,x) = 1 + 7*x + 252*x^2 + 12516*x^3 + 737814*x^4 + 49006986*x^5 +...
%e F(7,x) = 1 + 8*x + 360*x^2 + 22200*x^3 + 1610760*x^4 + 130516488*x^5 +...
%e ...
%e F(1,x)^2 = 1 + 4*x + 28*x^2 + 280*x^3 + 3736*x^4 + 64084*x^5 +...
%e F(2,x)^3 = 1 + 9*x + 117*x^2 + 1917*x^3 + 38025*x^4 + 897417*x^5 +...
%e F(3,x)^4 = 1 + 16*x + 336*x^2 + 8336*x^3 + 236976*x^4 + 7626976*x^5 +...
%e F(4,x)^5 = 1 + 25*x + 775*x^2 + 27325*x^3 + 1068175*x^4 + 45826525*x^5 +...
%e F(5,x)^6 = 1 + 36*x + 1548*x^2 + 73872*x^3 + 3826656*x^4 + 213200856*x^5 +...
%e F(6,x)^7 = 1 + 49*x + 2793*x^2 + 173705*x^3 + 11558561*x^4 + 815959809*x^5 +...
%e F(7,x)^8 = 1 + 64*x + 4672*x^2 + 367552*x^3 + 30617920*x^4 + 2679625792*x^5 +...
%e ...
%o (PARI) {a(n)=local(A=1+n*x);for(k=0,n-1,A=(1 - (n-k)^2*x*A +x*O(x^n))^(-1/(n-k)));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, May 31 2014
%Y Cf. A243195.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2002
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