A319083
Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 7, 17, 9, 1, 0, 6, 38, 39, 12, 1, 0, 12, 70, 120, 70, 15, 1, 0, 8, 116, 300, 280, 110, 18, 1, 0, 15, 185, 645, 885, 545, 159, 21, 1, 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1, 0, 18, 384, 2262, 5586, 6713, 4281, 1498, 284, 27, 1
Offset: 0
Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 3, 1;
[3] 0, 4, 6, 1;
[4] 0, 7, 17, 9, 1;
[5] 0, 6, 38, 39, 12, 1;
[6] 0, 12, 70, 120, 70, 15, 1;
[7] 0, 8, 116, 300, 280, 110, 18, 1;
[8] 0, 15, 185, 645, 885, 545, 159, 21, 1;
[9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;
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P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-sigma); # Peter Luschny, Oct 19 2022
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T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
If[k == 1, If[n == 0, 0, DivisorSigma[1, n]],
With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)
A374978
a(n) = Sum_{i+j+k+l+m=n, i,j,k,l,m >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m).
Original entry on oeis.org
0, 0, 0, 0, 1, 15, 110, 545, 2095, 6713, 18750, 47040, 108185, 231640, 467034, 894605, 1639680, 2891475, 4929660, 8155182, 13135080, 20651875, 31770970, 47923680, 70989801, 103454645, 148464520, 210155730, 293558265, 405325092, 553175000, 747508125, 999747750
Offset: 1
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b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 5):
seq(a(n), n=1..55); # Alois P. Heinz, Jul 26 2024
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b[n_, k_] := b[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, DivisorSigma[1, n]], Function[q, Sum[b[j, q]*b[n - j, k - q], {j, 0, n}]][Quotient[k, 2]]]];
a[n_] := b[n, 5];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jul 11 2025, after Alois P. Heinz *)
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from sympy import divisor_sigma
def A374978(n): return sum(divisor_sigma(j)*sum((5*divisor_sigma(i+1,3)-(5+6*i)*divisor_sigma(i+1))*(5*divisor_sigma(n-j-i-1,3)-(5+6*(n-j-i-2))*divisor_sigma(n-j-i-1)) for i in range(1,n-j-2)) for j in range(1,n))//144
A374979
a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m)*sigma(r).
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 18, 159, 942, 4281, 16050, 51932, 149532, 391524, 947246, 2143677, 4581204, 9316195, 18138636, 33984912, 61534652, 108055425, 184582014, 307515038, 500798058, 798762453, 1249917936, 1921788036, 2907159804, 4332046200, 6365441400, 9232216725
Offset: 1
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b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 6):
seq(a(n), n=1..55); # Alois P. Heinz, Jul 26 2024
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from functools import lru_cache
from sympy import divisor_sigma
def A374979(n):
@lru_cache(maxsize=None)
def g(x):
f = factorint(x+1).items()
return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*x)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12
return sum(g(i)*g(j)*g(n-3-i-j) for i in range(1,n-4) for j in range(1,n-i-3))
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