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 A179500 G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k>=0} a(k)^n* x^k]^n* x^n/n ). 2

%I #3 Mar 30 2012 18:37:22

%S 1,1,2,5,16,74,612,12271,893422,414194958,2790004382642,

%T 907459561737399050,79479770316224310083608800,

%U 22570656733849188237806831031463922346

%N G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k>=0} a(k)^n* x^k]^n* x^n/n ).

%H Paul D. Hanna, <a href="/A179500/b179500.txt">Table of n, a(n), n = 0..20.</a>

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 74*x^5 + 612*x^6 +...

%e The logarithm (A179501) begins:

%e log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 276*x^5/5 + 3138*x^6/6 + 80998*x^7/7 + 7043187*x^8/8 + 3719589796*x^9/9 +...

%e and equals the series:

%e log(A(x)) = (1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 74*x^5 +...)*x

%e + (1 + x + 2^2*x^2 + 5^2*x^3 + 16^2*x^4 + 74^2*x^5 +...)^2*x^2/2

%e + (1 + x + 2^3*x^2 + 5^3*x^3 + 16^3*x^4 + 74^3*x^5 +...)^3*x^3/3

%e + (1 + x + 2^4*x^2 + 5^4*x^3 + 16^4*x^4 + 74^4*x^5 +...)^4*x^4/4

%e + (1 + x + 2^5*x^2 + 5^5*x^3 + 16^5*x^4 + 74^5*x^5 +...)^5*x^5/5 +...

%e More explicitly,

%e log(A(x)) = (1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 74*x^5 +...)*x

%e + (1 + 2*x + 9*x^2 + 58*x^3 + 578*x^4 + 11664*x^5 +...)*x^2/2

%e + (1 + 3*x + 27*x^2 + 424*x^3 + 13254*x^4 +...)*x^3/3

%e + (1 + 4*x + 70*x^2 + 2696*x^3 + 271373*x^4 +...)*x^4/4

%e + (1 + 5*x + 170*x^2 + 16275*x^3 + 5316585*x^4 +...)*x^5/5 +...

%o (PARI) {a(n)=local(A);A=exp(sum(m=1,n,sum(k=0,n-m,a(k)^m*x^k+x*O(x^n))^m*x^m/m));if(n==0,1,polcoeff(A,n))}

%Y Cf. A179501.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 21 2010

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Last modified August 12 19:26 EDT 2024. Contains 375113 sequences. (Running on oeis4.)