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%I #19 Sep 17 2017 18:21:00
%S 1,1,3,10,41,219,1602,16635,247171,5242108,157390565,6663089873,
%T 396778864166,33200932308437,3906922702271961,646161881511137940,
%U 150482521507292513413,49318093291540113084965,22790150225552744270503692,14843990673285561887923674163,13646527810852572644275538963207,17710656073227095563348293151121448
%N G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^n*x^k]^n/n ), a power series in x with integer coefficients.
%H Paul D. Hanna, <a href="/A156170/b156170.txt">Table of n, a(n) for n = 0..150</a>
%F G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * x^k / (1-x)^(n+1) ]^n / n ), where A008292 are the Eulerian numbers. - _Paul D. Hanna_, Sep 13 2016
%F Conjecture: log(a(n)) ~ n^2 * log(2)/4. - _Vaclav Kotesovec_, Sep 02 2017
%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 41*x^4 + 219*x^5 + 1602*x^6 +...
%e log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 117*x^4/4 + 821*x^5/5 + 7796*x^6/6 + 1810093*x^7/7 + 44561794*x^8/8 +...+ A276750(n)*x^n/n +...
%e The logarithm of g.f. A(x) equals the series:
%e log(A(x)) = Sum_{n>=1} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n/n,
%e or,
%e log(A(x)) = (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 +...) +
%e (x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + 5^2*x^5 +...)^2/2 +
%e (x + 2^3*x^2 + 3^3*x^3 + 4^3*x^4 + 5^3*x^5 +...)^3/3 +
%e (x + 2^4*x^2 + 3^4*x^3 + 4^4*x^4 + 5^4*x^5 +...)^4/4 + ...
%e This logarithmic series can be written using the Eulerian numbers like so:
%e log(A(x)) = x/(1-x)^2 + (x + x^2)^2/(1-x)^6/2 + (x + 4*x^2 + x^3)^3/(1-x)^12/3 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20/4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30/5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n^2+n)/n +...
%o (PARI) {a(n) = polcoeff( exp( sum(m=1,n, sum(k=1,n, k^m*x^k +x*O(x^n))^m/m ) ),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {A008292(n,k) = sum(j=0,k, (-1)^j * (k-j)^n * binomial(n+1,j))}
%o {a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1,n+1, sum(k=1,m, A008292(m,k)*x^k/(1-x +Oxn)^(m+1) )^m / m ) ); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A276750, A276751, A276752, A156171, A155200, A008292, A292500.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 05 2009
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