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

1, -3, 28, -707, 44576, -6695766, 2324916672, -1827405843363, 3201960457239040, -12360766887879809858, 104155804285010077051904, -1901288103716422362163490318, 74716157012390526276910403768320, -6287554722733254962685763077329845772, 1127914878748595440161120152354758317867008

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

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.

Example

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.
The g.f. can be written using the Eulerian numbers like so:
-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)) +...
The terms also form the logarithm of an integer series:
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 +...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
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]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A276744, A008292.

Sequence in context: A181588 A084880 A110259 * A354664 A015474 A324462

Adjacent sequences: A276742 A276743 A276744 * A276746 A276747 A276748

Keyword

sign

Author

Paul D. Hanna, Sep 29 2016

Status

approved