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A125282 G.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x).

%I A125282 #12 Feb 24 2014 06:45:44
%S A125282 1,1,2,5,17,80,525,4839,62936,1158785,30277579,1124649526,59465788597,
%T A125282 4480380804517,481401971074410,73812092299235769,16158739669470307453,
%U A125282 5052972095683109687920,2257981256268589345121153
%N A125282 G.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x).
%H A125282 Vaclav Kotesovec, <a href="/A125282/b125282.txt">Table of n, a(n) for n = 0..100</a>
%H A125282 Vaclav Kotesovec, <a href="/A125282/a125282.jpg">Graph, asymptotic behavior</a>
%F A125282 a(n) = Sum_{k=0..n-1} (n-k)^k * a(k) for n>0 with a(0)=1.
%F A125282 Limit n->infinity (a(n))^(1/n^2) = 3^(1/6). - _Vaclav Kotesovec_, Feb 24 2014
%F A125282 a(n) ~ c * 3^(n^2/6 - n/2), where c = 372374.41350200494715367264093778... if n=3k, c = 372374.41350258936507380006951913... if n=3k+1, and c = 372374.41350254286383864609841301... if n=3k+2. - _Vaclav Kotesovec_, Feb 24 2014
%e A125282 A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 80*x^5 + 525*x^6 + 4839*x^7 +...
%e A125282 G.f. A(x) satisfies:
%e A125282 A(x) = 1 + x*A(x) + x^2*A(2x) + x^3*A(3x) + x^4*A(4x) + x^5*A(5x) +...
%e A125282 which leads to the recurrence illustrated by:
%e A125282 a(4) = 4^0*(1) + 3^1*(1) + 2^2*(2) + 1^3*(5) = 17;
%e A125282 a(5) = 5^0*(1) + 4^1*(1) + 3^2*(2) + 2^3*(5) + 1^4*(17) = 80;
%e A125282 a(6) = 6^0*(1) + 5^1*(1) + 4^2*(2) + 3^3*(5) + 2^4*(17) + 1^5*(80) = 525.
%t A125282 nmax = 20; aa = ConstantArray[0, nmax]; aa[[1]] = 1;Do[aa[[n]] = 1 + Sum[(n-k)^k*aa[[k]], {k, 1, n - 1}], {n, 2, nmax}]; Flatten[{1, aa}] (* _Vaclav Kotesovec_, Feb 23 2014 *)
%o A125282 (PARI) {a(n)=if(n==0,1,sum(k=0,n-1,(n-k)^k*a(k)))}
%Y A125282 Cf. A125281 (variant), A210525.
%K A125282 nonn
%O A125282 0,3
%A A125282 _Paul D. Hanna_, Nov 29 2006