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%I #9 Jun 01 2014 20:20:15
%S 1,1,1,3,5,13,61,133,449,1825,11497,29905,121529,613121,3192553,
%T 26963653,76748369,367110269,2101537105,13742608029,90490605353,
%U 966603833657,2899027074937,15202727310033,102739122225929,738145449190921,6064931145859705,47996143247509851,637518525737986877
%N G.f. A(x) = F(0,x) where F(0,x) = 1/(1 - x*F(1,x)), F(1,x) = 1/(1 - (2*x*F(2,x))^2)^(1/2), F(2,x) = 1/(1 - (3*x*F(3,x))^3)^(1/3), ..., so that F(n-1,x)^n = 1/(1 - (n*x*F(n,x))^n) for n>=0.
%H Paul D. Hanna, <a href="/A243161/b243161.txt">Table of n, a(n) for n = 0..250</a>
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 13*x^5 + 61*x^6 + 133*x^7 + 449*x^8 + 1825*x^9 + 11497*x^10 + 29905*x^11 + 121529*x^12 +...
%e Define F(n,x) with F(0,x) = A(x), where for n>0 we have
%e F(n-1,x) = 1/(1 - (n*x*F(n,x))^n )^(1/n), or equivalently
%e F(n,x) = (1 - 1/F(n-1,x)^n)^(1/n) / (n*x),
%e then
%e F(1,x) = 1 + 2*x^2 + 3*2*x^4 + 6^2*x^5 + 5*2^2*x^6 + 3^3*2^3*x^7 +...
%e F(2,x) = 1 + 3^2*x^3 + 6*3^3*x^6 + 12^3*x^7 + 14*3^5*x^9 + 4^4*3^5*x^10 +...
%e F(3,x) = 1 + 4^3*x^4 + 10*4^5*x^8 + 20^4*x^9 + 30*4^8*x^12 + 5^5*4^7*x^13 +...
%e F(4,x) = 1 + 5^4*x^5 + 15*5^7*x^10 + 30^5*x^11 + 55*5^11*x^15 + 6^6*5^9*x^16 +...
%e F(5,x) = 1 + 6^5*x^6 + 21*6^9*x^12 + 42^6*x^13 + 91*6^14*x^18 + 7^7*6^11*x^19 +...
%e ....
%e F(1,x)^2 = 1 + 2^2*x^2 + 2^4*x^4 + 2^3*3^2*x^5 + 2^6*x^6 + 2*3^2*2^5*x^7 +...
%e F(2,x)^3 = 1 + 3^3*x^3 + 3^6*x^6 + 3^4*4^3*x^7 + 3^9*x^9 + 2*4^3*3^7*x^10 +...
%e F(3,x)^4 = 1 + 4^4*x^4 + 4^8*x^8 + 4^5*5^4*x^9 + 4^12*x^12 + 2*5^4*4^9*x^13 +...
%e F(4,x)^5 = 1 + 5^5*x^5 + 5^10*x^10 + 5^6*6^5*x^11 + 5^15*x^15 + 2*6^5*5^11*x^16 +...
%e F(5,x)^6 = 1 + 6^6*x^6 + 6^12*x^12 + 6^7*7^6*x^13 + 6^18*x^18 + 2*7^6*6^13*x^19 +...
%e ...
%o (PARI) {a(n)=local(A=1+n*x); for(k=0, n-1, A=(1 - ((n-k)*x*A)^(n-k) +x*O(x^n))^(-1/(n-k))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A243195, A075820.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jun 01 2014
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