A206766 - OEIS

%I #12 Oct 25 2024 04:50:25

%S 1,23,226,3039,33306,594902,10012010,220553599,5170061143,

%T 138942811678,4049569009674,130045043225838,4503599691290714,

%U 168477832912220134,6746676272050878036,288487396687082933759,13107200000016921588858,630907565930072760920429

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

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

%H Vaclav Kotesovec, <a href="/A206766/b206766.txt">Table of n, a(n) for n = 1..380</a>

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

%F Logarithmic derivative of A206765.

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

%F a(n) ~ exp(3) * n^n. - _Vaclav Kotesovec_, Oct 04 2020

%e L.g.f.: L(x) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...

%e Exponentiation yields the g.f. of A206765:

%e exp(L(x)) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...

%e Illustration of terms.

%e a(2) = 2*sigma(2,1)*3 + 1*sigma(2,2)*1 = 2*3*3 + 1*5*1 = 23;

%e 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;

%e 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.

%o (PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*3^(n-k))}

%o (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)}

%o for(n=1,21,print1(a(n),", "))

%Y Cf. A206765 (exp), A205815, A205812.

%K nonn,easy

%O 1,2

%A _Paul D. Hanna_, Feb 12 2012