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A276745 G.f.: -log(1-x) = Sum_{n>=1} a(n) * [ Sum_{k>=1} k^n * x^k ]^n / n. 2

%I

%S 1,-3,28,-707,44576,-6695766,2324916672,-1827405843363,

%T 3201960457239040,-12360766887879809858,104155804285010077051904,

%U -1901288103716422362163490318,74716157012390526276910403768320,-6287554722733254962685763077329845772,1127914878748595440161120152354758317867008

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

%H Paul D. Hanna, <a href="/A276745/b276745.txt">Table of n, a(n) for n = 1..100</a>

%F G.f.: -log(1-x) = Sum_{n>=1} a(n) * [ Sum_{k=1..n} A008292(n,k) * x^k ]^n/n / (1-x)^(n*(n+1)), where A008292 are the Eulerian numbers.

%e G.f.: -log(1-x) = Sum_{n>=1} a(n) * (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n/n.

%e The g.f. can be written using the Eulerian numbers like so:

%e -log(1-x) = x/(1-x)^2 - 3*(x + x^2)^2/(1-x)^6/2 + 28*(x + 4*x^2 + x^3)^3/(1-x)^12/3 - 707*(x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20/4 + 44576*(x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30/5 - 6695766*(x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42/6 +...+ a(n)*[ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)) +...

%e The terms also form the logarithm of an integer series:

%e exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + x - x^2 + 8*x^3 - 167*x^4 + 8730*x^5 - 1106838*x^6 + 331004710*x^7 - 228092398323*x^8 + 355544615876962*x^9 - 1235720684096631440*x^10 + 9467473046100717088970*x^11 - 158431205361256873139914550*x^12 +...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = 1 - (#A)*Vec(sum(m=1,#A,A[m]*sum(k=1,#A+1,k^m * x^k +x*O(x^#A))^m/m))[#A] ); A[n]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A276744, A008292.

%K sign

%O 1,2

%A _Paul D. Hanna_, Sep 29 2016

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Last modified August 20 19:51 EDT 2019. Contains 326155 sequences. (Running on oeis4.)