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A162656 E.g.f. satisfies: A(x) = (1 + x*A(x))^(A(x)^2).

%I A162656 #14 Nov 22 2021 02:37:59
%S A162656 1,1,6,78,1564,42660,1475028,61838000,3048628336,172850367744,
%T A162656 11081419939680,792685606699008,62593631986243488,5408185369666834560,
%U A162656 507526235156395322112,51407696054804232576000
%N A162656 E.g.f. satisfies: A(x) = (1 + x*A(x))^(A(x)^2).
%H A162656 Seiichi Manyama, <a href="/A162656/b162656.txt">Table of n, a(n) for n = 0..337</a>
%F A162656 (1) a(n) = Sum_{k=0..n} (n+2k+1)^(k-1) * Stirling1(n,k).
%F A162656 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F A162656 (2) a(n,m) = Sum_{k=0..n} m*(n+2k+m)^(k-1) * Stirling1(n,k) ;
%F A162656 which is equivalent to the following:
%F A162656 (3) a(n,m) = Sum_{k=0..n} m*(n+2k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1-j*x) };
%F A162656 (4) a(n,m) = n!*Sum_{k=0..n} m*(n+2k+m)^(k-1) * {[x^(n-k)] (log(1+x)/x)^k/k!}.
%F A162656 a(n) ~ s^2*sqrt(r*s*(1+r*s) / (2+r*s*(4+3*s^2+2*r*s*(1+s^2)))) * n^(n-1) / (exp(n)*r^n), where r = 0.1337990328596883051... and s = 1.408676495644569372... are roots of the system of equations s^2*(r*s + 2*(1+r*s)*log(1+r*s)) = 1+r*s, (1+r*s)^(s^2) = s. - _Vaclav Kotesovec_, Jul 15 2014
%e A162656 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1564*x^4/4! + 42660*x^5/5! +...
%e A162656 log(A(x)) = A(x)^2*log(1 + x*A(x)) where
%e A162656 log(A(x)) = x + 5*x^2/2! + 62*x^3/3! + 1210*x^4/4! + 32464*x^5/5! +...
%e A162656 log(1 + x*A(x)) = x + x^2/2! + 14*x^3/3! + 246*x^4/4! + 6284*x^5/5! +...
%t A162656 Table[Sum[(n+2*k+1)^(k-1) * StirlingS1[n,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 15 2014 *)
%o A162656 (PARI) {a(n,m=1)=sum(k=0,n,m*(n+2*k+m)^(k-1)*polcoeff(prod(j=1,n-1,1-j*x),n-k))}
%o A162656 (PARI) {a(n,m=1)=sum(k=0,n,m*(n+2*k+m)^(k-1)*n!/k!*polcoeff((log(1+x+x*O(x^n))/x)^k,n-k))}
%o A162656 (PARI) a(n,m=1)=sum(k=0,n,m*(n+2*k+m)^(k-1)*stirling(n, k, 1));
%Y A162656 Cf. A008275 (Stirling1), variants: A162655, A141209.
%K A162656 nonn
%O A162656 0,3
%A A162656 _Paul D. Hanna_, Jul 08 2009