cp's OEIS Frontend

A278403 a(n) = Sum_{d|n} d^2 * (d+1)/2.

%I A278403 #27 Jan 02 2025 07:53:46
%S A278403 1,7,19,47,76,151,197,335,424,632,727,1127,1184,1673,1894,2511,2602,
%T A278403 3634,3611,4872,5066,6299,6349,8615,8201,10316,10630,13081,12616,
%U A278403 16526,15377,19407,19258,22838,22322,28586,26012,31775,31622,37960,35302,44594,40679,49899,48874,56081,53017,67239,60222,72507,70246,82012,75844,94030,85502,102745,97850,111860,104431,131502
%N A278403 a(n) = Sum_{d|n} d^2 * (d+1)/2.
%H A278403 Paul D. Hanna, <a href="/A278403/b278403.txt">Table of n, a(n) for n = 1..1000</a>
%F A278403 Let the l.g.f. be L(x) = Sum_{n>=1} a(n)*x^n/n, then:
%F A278403 (1) exp( L(x) ) = Product_{n>=1} 1/(1 - x^n)^(n*(n+1)/2),
%F A278403 (2) L(x) =  Sum_{n>=1} (x^n/n) / (1 - x^n)^3.
%F A278403 O.g.f.: Sum_{n>=1} n^2*(n+1)/2 * x^n / (1 - x^n).
%F A278403 a(n) = (sigma_3(n) + sigma_2(n))/2, where sigma_2(n) = A001157(n) and sigma_3(n) = A001158(n).
%F A278403 Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 720. - _Vaclav Kotesovec_, Jul 13 2021
%F A278403 Dirichlet g.f.: zeta(s) * (zeta(s-3) + zeta(s-2)) / 2. - _Amiram Eldar_, Jan 02 2025
%e A278403 L.g.f.: L(x) = x + 7*x^2/2 + 19*x^3/3 + 47*x^4/4 + 76*x^5/5 + 151*x^6/6 + 197*x^7/7 + 335*x^8/8 + 424*x^9/9 + 632*x^10/10 + 727*x^11/11 + 1127*x^12/12 +...
%e A278403 which equals the series
%e A278403 L(x) = x/(1-x)^3 + (x^2/2)/(1-x^2)^3 + (x^3/3)/(1-x^3)^3 + (x^4/4)/(1-x^4)^3 + (x^5/5)/(1-x^5)^3 + (x^6/6)/(1-x^6)^3 + (x^7/7)/(1-x^7)^3 +...
%e A278403 The exponentiation of the l.g.f. equals the infinite product
%e A278403 exp(L(x)) = 1/((1-x)*(1-x^2)^3*(1-x^3)^6*(1-x^4)^10*(1-x^5)^15*(1-x^6)^21*...);
%e A278403 explicitly,
%e A278403 exp(L(x)) = 1 + x + 4*x^2 + 10*x^3 + 26*x^4 + 59*x^5 + 141*x^6 + 310*x^7 + 692*x^8 + 1483*x^9 + 3162*x^10 + 6583*x^11 + 13602*x^12 +...+ A000294(n)*x^n +...
%t A278403 Table[Total[#^2*(#+1)/2&/@Divisors[n]],{n,60}] (* _Harvey P. Dale_, Jul 26 2017 *)
%o A278403 (PARI) {a(n) = sumdiv(n,d,d^2*(d+1)/2)}
%o A278403 for(n=1,60,print1(a(n),", "))
%o A278403 (PARI) {a(n) = (sigma(n,3) + sigma(n,2))/2}
%o A278403 for(n=1,60,print1(a(n),", "))
%o A278403 (PARI) {a(n) = n * polcoeff( sum(k=1,n, (x^k/k) / (1 - x^k +x*O(x^n))^3), n)}
%o A278403 for(n=1,60,print1(a(n),", "))
%Y A278403 Cf. A000294, A001157, A001158, A277613.
%K A278403 nonn,easy
%O A278403 1,2
%A A278403 _Paul D. Hanna_, Nov 20 2016