A206766 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 3^(n-k).

1, 23, 226, 3039, 33306, 594902, 10012010, 220553599, 5170061143, 138942811678, 4049569009674, 130045043225838, 4503599691290714, 168477832912220134, 6746676272050878036, 288487396687082933759, 13107200000016921588858, 630907565930072760920429

Offset

1,2

Comments

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..380

Formula

a(n) = Sum_{d|n} ((d+3)^n - 3^n).

Logarithmic derivative of A206765.

L.g.f.: Sum_{n>=1} (1/n) * log( (1 - 3^n*x^n) / (1 - (n+3)^n*x^n) ).

a(n) ~ exp(3) * n^n. - Vaclav Kotesovec, Oct 04 2020

Example

L.g.f.: L(x) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...
Exponentiation yields the g.f. of A206765:
exp(L(x) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1)*3 + 1*sigma(2,2)*1 = 2*3*3 + 1*5*1 = 23;
a(3) = 3*sigma(3,1)*9 + 3*sigma(3,2)*3 + 1*sigma(3,3)*1 = 3*4*9 + 3*10*3 + 1*28*1 = 226;
a(4) = 4*sigma(4,1)*27 + 6*sigma(4,2)*9 + 4*sigma(4,3)*3 + 1*sigma(4,4)*1 = 4*7*27 + 6*21*9 + 4*73*3 + 1*273*1 = 3039.

Prog

(PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*3^(n-k))}
(PARI) {a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-3^k*x^k)/(1-(k+3)^k*x^k +x*O(x^n)))), n)}
for(n=1, 21, print1(a(n), ", "))

Crossrefs

Cf. A206765 (exp), A205815, A205812.

Sequence in context: A126544 A272796 A285552 * A142490 A051819 A179631

Adjacent sequences: A206763 A206764 A206765 * A206767 A206768 A206769

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2012

Status

approved