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%I #12 Aug 30 2022 14:14:56
%S 1,2,5,26,501,42262,14564184,18926665052,96371663657380,
%T 1825266130738144920,136764680697906838980633,
%U 38133043109557952095731186822,42464330390232136488003531922964743
%N G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} binomial(n,k)^n.
%H G. C. Greubel, <a href="/A167007/b167007.txt">Table of n, a(n) for n = 0..59</a>
%F a(n) = (1/n)*Sum_{k=1..n} A167010(k)*a(n-k) for n>0 with a(0)=1. - _Paul D. Hanna_, Nov 25 2009
%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 501*x^4 + 42262*x^5 + ...
%e log(A(x)) = 2*x + 6*x^2/2 + 56*x^3/3 + 1810*x^4/4 + 206252*x^5/5 + 86874564*x^6/6 + ... + A167010(n)*x^n/n + ...
%t A167010[n_]:= A167010[n]= Sum[Binomial[n,j]^n, {j,0,n}];
%t A167007[n_]:= A167007[n]= If[n==0, 1, (1/n)*Sum[A167010[j]*A167007[n-j], {j,n}]];
%t Table[A167007[n], {n,0,30}] (* _G. C. Greubel_, Aug 26 2022 *)
%o (PARI) {a(n) = polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^m)*x^m/m) +x*O(x^n)), n)};
%o (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sum(j=0, k, binomial(k, j)^k)*a(n-k)))} \\ _Paul D. Hanna_, Nov 25 2009
%o (Magma)
%o A167010:= func< n | (&+[Binomial(n,j)^n: j in [0..n]]) >;
%o function A167007(n)
%o if n lt 2 then return n+1;
%o else return (&+[A167010(j)*A167007(n-j): j in [1..n]])/n;
%o end if; return A167007;
%o end function;
%o [A167007(n): n in [0..20]]; // _G. C. Greubel_, Aug 26 2022
%o (SageMath)
%o def A167010(n): return sum(binomial(n,j)^n for j in (0..n))
%o def A167007(n): return 1 if (n==0) else (1/n)*sum( A167010(j)*A167007(n-j) for j in (1..n))
%o [A167007(n) for n in (0..30)] # _G. C. Greubel_, Aug 26 2022
%Y Cf. A155200, A167006, A167010.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 17 2009
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