A192890 Expansion of g.f.: exp( Sum_{n>=1} 2^n*(Sum_{d|n} d*x^d)^n/n ).

1, 2, 4, 16, 40, 136, 400, 1256, 4272, 14984, 51856, 174304, 563776, 1820672, 5780864, 18518912, 61112832, 213235904, 775157504, 2859529984, 10441532416, 37338042528, 130397329216, 447932053120, 1524595536512, 5194654080416, 17800611666496

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 40*x^4 + 136*x^5 + 400*x^6 + 1256*x^7 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 4*(x + 2*x^2)^2/2 + 8*(x + 3*x^3)^3/3 + 16*(x + 2*x^2 + 4*x^4)^4/4 + 32*(x + 5*x^5)^5/5 + 64*(x + 2*x^2 + 3*x^3 + 6*x^6)^6/6 + 128*(x + 7*x^7)^7/7 + 256*(x + 2*x^2 + 4*x^4 + 8*x^8)^8/8 +...
Explicitly, the logarithmic series begins:
log(A(x)) = 2*x + 4*x^2/2 + 32*x^3/3 + 48*x^4/4 + 312*x^5/5 + 640*x^6/6 + 2872*x^7/7 + 10496*x^8/8  + 46760*x^9/9 + 162624*x^10/10 +...

Mathematica

With[{m = 30}, CoefficientList[Series[Exp[Sum[2^n (Sum[d*x^d, {d, Divisors[n]}])^n/n, {n, 1, m + 2}]], {x, 0, m}], x]] (* G. C. Greubel, Jan 09 2019 *)

Prog

(PARI) {a(n)=local(A); A=exp(sum(m=1, n+1, 2^m*sumdiv(m, d, d*x^d +x*O(x^n))^m/m)); polcoeff(A, n)}

Crossrefs

Cf. A192860, A192891.

Sequence in context: A223093 A333022 A048222 * A382299 A062330 A133465

Adjacent sequences: A192887 A192888 A192889 * A192891 A192892 A192893

Keyword

nonn

Author

Paul D. Hanna, Jul 11 2011

Status

approved