A209803 a(n) = Sum_{d|n} d*2^(n*d).

2, 36, 1544, 262672, 167772192, 412317655104, 3940649673949312, 147573952606856413440, 21760664753063325547364864, 12676506002282299644466568365056, 29243015907268149203883755326167582720, 267608942382367477698456953471803662041878528

Offset

1,1

Links

Table of n, a(n) for n=1..12.

Formula

L.g.f.: Sum_{n>=1} -log(1 - 2^(n^2)*x^n).

L.g.f.: log( Sum_{n>=0} 2^(n^2)*x^n / Product_{k=1..n} (1 - 2^(k^2)*x^k) ).

Equals the logarithmic derivative of A157317.

Example

L.g.f.: L(x) = 2*x + 36*x^2/2 + 1544*x^3/3 + 262672*x^4/4 + ...
where exponentiation yields the g.f. of A157317:
exp(L(x)) = 1 + 2*x + 20*x^2 + 552*x^3 + 66896*x^4 + 33696416*x^5 + ...

Mathematica

a[n_] := DivisorSum[n, # * 2^(n*#) &]; Array[a, 12] (* Amiram Eldar, Aug 22 2023 *)

Prog

(PARI) {a(n)=sumdiv(n, d, d*2^(n*d))}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, -log(1 - 2^(m^2)*x^m +x*O(x^n))), n)}
(PARI) {a(n)=n*polcoeff(log(1+sum(m=1, n, 2^(m^2)*x^m/prod(k=1, m, 1-(2^k*x)^k+x*O(x^n)))), n)}
for(n=1, 15, print1(a(n), ", "))

Crossrefs

Cf. A157317, A209804.

Sequence in context: A046673 A245959 A174580 * A088026 A174881 A126934

Adjacent sequences: A209800 A209801 A209802 * A209804 A209805 A209806

Keyword

nonn

Author

Paul D. Hanna, Mar 13 2012

Status

approved