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A049075 Eigensequence of a power series transformation.

%I A049075 #20 Dec 19 2014 20:36:19
%S A049075 1,1,2,4,8,18,43,102,247,617,1564,4003,10355,27051,71225,188743,
%T A049075 503111,1348301,3630294,9815159,26637436,72540432,198162708,542875096,
%U A049075 1491126550,4105602719,11329408543,31328137525,86795258650,240898943969,669730499207,1864855943748
%N A049075 Eigensequence of a power series transformation.
%C A049075 Euler transform of a(n) - if( n%4, 0, a(n/2)) is sequence itself with offset 0.
%H A049075 Alois P. Heinz, <a href="/A049075/b049075.txt">Table of n, a(n) for n = 1..650</a>
%F A049075 G.f.: A(x) = x exp(A(x) - A(-x^2)/2 + A(x^3)/3 - A(-x^4)/4 + ...). Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+(-x)^n)^((-1)^n*a(n)).
%F A049075 G.f.: x prod_{n>0} (1-x^(4n))^a(2n)/(1-x^n)^a(n).
%F A049075 a(n) ~ c * d^n / n^(3/2), where d = 2.92045137601697174071599643..., c = 0.4299447159290328896620383... . - _Vaclav Kotesovec_, Aug 25 2014
%e A049075 x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 43*x^7 + 102*x^8 + 247*x^9 + 617*x^10 + ...
%p A049075 with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(n-> a(n) -`if`(modp(n,4)<>0, 0,a(n/2))): a:= n-> b(n-1): seq(a(n), n=1..40);  # _Alois P. Heinz_, Sep 06 2008
%t A049075 s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ](-1)^k ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ]
%o A049075 (PARI) {a(n) = local(A=x); if( n<1, 0, for( k=1, n-1, A *= (1 + (-x)^k + x*O(x^n))^((-1)^k * polcoeff(A, k))); polcoeff(A, n))}
%Y A049075 Cf. A045648, A000081, A004111.
%K A049075 nonn,eigen
%O A049075 1,3
%A A049075 _Michael Somos_, Aug 08 1999