A192859 L.g.f.: Sum_{n>=1} (Sum_{d|n} d*x^d)^n/n = Sum_{n>=1} a(n)*x^n/n.

1, 1, 7, 9, 26, 37, 162, 329, 1087, 2526, 6810, 15009, 32514, 66312, 136877, 276137, 632401, 1546003, 4648446, 14647994, 47229840, 145042888, 423449873, 1161865841, 3026208201, 7457926620, 17580700762, 39758132780, 87567965436, 188225902377

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

Equals the logarithmic derivative of A192860.

Example

L.g.f.: L(x) =  = x + x^2/2 + 7*x^3/3 + 9*x^4/4 + 26*x^5/5 + 37*x^6/6 +...
which is generated by:
L(x) = x + (x + 2*x^2)^2/2 + (x + 3*x^3)^3/3 + (x + 2*x^2 + 4*x^4)^4/4 + (x + 5*x^5)^5/5 + (x + 2*x^2 + 3*x^3 + 6*x^6)^6/6 + (x + 7*x^7)^7/7 + (x + 2*x^2 + 4*x^4 + 8*x^8)^8/8 +...
Exponentiation of the l.g.f. yields the g.f. of A192860:
exp(L(x)) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 45*x^7 +...

Mathematica

With[{m = 30}, Rest[CoefficientList[Series[Sum[(Sum[d*x^d, {d, Divisors[n] }])^n/n, {n, 1, m + 2}], {x, 0, m}], x]]*Range[1, m]] (* G. C. Greubel, Jan 07 2019 *)

Prog

(PARI) {a(n)=local(A=x); A=sum(m=1, n+1, sumdiv(m, d, d*x^d +x*O(x^n))^m/m); n*polcoeff(A, n)}

Crossrefs

Cf. A192860.

Sequence in context: A102028 A265990 A125260 * A104703 A032617 A113124

Adjacent sequences: A192856 A192857 A192858 * A192860 A192861 A192862

Keyword

nonn

Author

Paul D. Hanna, Jul 11 2011

Status

approved