cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A374951 a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).

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%I A374951 #35 Mar 14 2025 06:15:08
%S A374951 0,0,1,9,39,120,300,645,1261,2262,3825,6160,9471,14178,20376,28965,
%T A374951 39600,54066,71145,94248,120140,155310,193116,244560,297819,370860,
%U A374951 443710,544554,641655,778458,904800,1085445,1248762,1483308,1688052,1991515,2244375,2626380
%N A374951 a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).
%H A374951 Vaclav Kotesovec, <a href="/A374951/b374951.txt">Table of n, a(n) for n = 1..10000</a>
%F A374951 G.f.: ( Sum_{k>=1} k * x^k/(1 - x^k) )^3 = ( Sum_{k>=1} x^k/(1 - x^k)^2 )^3.
%F A374951 a(n) = Sum_{i=1..n-2} sigma(i)*A000385(n-i-1). - _Chai Wah Wu_, Jul 25 2024
%F A374951 Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 155520. - _Vaclav Kotesovec_, Sep 19 2024
%p A374951 b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A374951       `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
%p A374951        add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A374951     end:
%p A374951 a:= n-> b(n, 3):
%p A374951 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jul 25 2024
%t A374951 b[n_, k_] := b[n, k] = If[k == 0, If[n == 0, 1, 0],
%t A374951    If[k == 1, If[n == 0, 0, DivisorSigma[1, n]], Function[q,
%t A374951    Sum[b[j, q]*b[n - j, k - q], {j, 0, n}]][Quotient[k, 2]]]];
%t A374951 a[n_] := b[n, 3];
%t A374951 Table[a[n], {n, 1, 55}] (* _Jean-François Alcover_, Mar 14 2025, after _Alois P. Heinz_ *)
%o A374951 (PARI) my(N=40, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k*x^k/(1-x^k))^3))
%o A374951 (Python)
%o A374951 from sympy import divisor_sigma
%o A374951 def A374951(n): return (60*sum(divisor_sigma(i)*divisor_sigma(n-i,3) for i in range(1,n))+divisor_sigma(n)*(9*n*(2*n-1)+1)-5*divisor_sigma(n,3)*(3*n-1))//144  # _Chai Wah Wu_, Jul 25 2024
%Y A374951 Column k=3 of A319083.
%Y A374951 Cf. A000203, A000385, A191829.
%K A374951 nonn
%O A374951 1,4
%A A374951 _Seiichi Manyama_, Jul 25 2024