A365667 -id:A365667 - cp's OEIS frontend

A060047 Triangle of generalized sum of divisors function, read by rows.

1, 2, 4, 1, 4, 2, 6, 4, 8, 8, 8, 14, 8, 1, 18, 13, 2, 28, 12, 4, 40, 12, 8, 52, 16, 14, 70, 14, 24, 88, 16, 40, 104, 24, 1, 56, 140, 16, 2, 84, 168, 18, 4, 122, 196, 26, 8, 168, 240, 20, 14, 232, 278, 24, 24, 312, 320, 32, 40, 408, 380, 24, 64, 528, 440, 24, 100, 672, 504

Offset: 1

N. J. A. Sloane, Mar 19 2001

Lengths of rows are 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 ... (A000196).

			Triangle turned on its side begins:
  1  2  4  4  6  8  8  8 13 12 12 ...
           1  2  4  8 14 18 28 40 ...
                          1  2  4 ...
For example, T(6,1) = 8, T(6,2) = 4.

G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A002131, A002132, A060046, A365666, A365667.

Cf. A060043, A060044, A060177, A060184.

T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*(2*m_1-1) + s_2*(2*m_2-1) + ... + s_k*(2*m_k-1) = n and the sum is over all such k-partitions of n.
G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(2*m_1+2*m_2+...+2*m_k-k)/((1-q^{2*m_1-1})*(1-q^{2*m_2-1})*...*(1-q^{2*m_k-1}))^2 = Sum_n T(n, k)*q^n.
G.f. for k-th diagonal: (-1)^k * (1/k) * ( Sum_{j>=k} (-1)^j * j * binomial(j+k-1,2*k-1) * q^(j^2) ) / ( 1 + 2 * Sum_{j>=1} (-q)^(j^2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto, Jan 24 2002

A365665 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 208, 390, 693, 1193, 1977, 3195, 4995, 7722, 11583, 17164, 24882, 35685, 50205, 70083, 96300, 131101, 176358, 235377, 310651, 407352, 529074, 682750, 874038, 1112085, 1405521, 1766259, 2206413, 2741431, 3389052, 4168089, 5103450, 6218469

Offset: 15

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with five designated summands (when part i has multiplicity j > 0 exactly one part i is "designated"). For example: a(16) = 3 because there are three partitions of 16 with five designated summands: [6'+ 4'+ 3'+ 2'+ 1'], [5'+ 4'+ 3'+ 2'+ 1'+ 1], [5'+ 4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 29 2025

Vaclav Kotesovec, Table of n, a(n) for n = 15..10000
G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365664.
Cf. A010816, A365631, A365667.
Column k=5 of A385001.
Cf. A384926.

nmax = 60; Drop[CoefficientList[Series[-1/11 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 5, 10]*x^(k*(k + 1)/2), {k, 5, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x], 15] (* Vaclav Kotesovec, Jul 29 2025 *)
(* or *)
Table[(10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080) * DivisorSigma[1, n] + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080) * DivisorSigma[3, n] + (133/1228800 - n/20480 + n^2/215040) * DivisorSigma[5, n] + (1/516096 - n/3096576) * DivisorSigma[7, n] + DivisorSigma[9, n]/154828800, {n, 15, 60}] (* Vaclav Kotesovec, Jul 29 2025 *)

G.f.: -(1/11) * ( Sum_{k>=5} (-1)^k * (2*k+1) * binomial(k+5,10) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
From Vaclav Kotesovec, Jul 29 2025: (Start)
a(n) = (10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080)*sigma(n) + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080)*sigma_3(n) + (133/1228800 - n/20480 + n^2/215040)*sigma_5(n) + (1/516096 - n/3096576)*sigma_7(n) + sigma_9(n)/154828800.
Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 144850083840000.
(End)

A365666 Expansion of Sum_{0

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 40, 64, 100, 144, 212, 304, 424, 588, 800, 1072, 1422, 1864, 2408, 3080, 3950, 4972, 6224, 7760, 9564, 11742, 14344, 17384, 20968, 25204, 30112, 35840, 42548, 50078, 58888, 69048, 80474, 93628, 108608, 125408, 144536, 166224, 190348

Offset: 16

Seiichi Manyama, Sep 15 2023

nonn

G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.

A diagonal of A060047.
Cf. A002131, A002132, A060046, A365667.
Cf. A015128.

nmax = 60; Drop[CoefficientList[Series[1/4 * Sum[(-1)^k*k*Binomial[k + 3, 7]*x^(k^2), {k, 4, nmax}]/(1 + 2*Sum[(-x)^(k^2), {k, 1, nmax}]), {x, 0, nmax}], x], 16] (* Vaclav Kotesovec, Jul 30 2025 *)

G.f.: (1/4) * ( Sum_{k>=4} (-1)^k * k * binomial(k+3,7) * q^(k^2) ) / ( 1 + 2 * Sum_{k>=1} (-q)^(k^2) ).

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A060047 Triangle of generalized sum of divisors function, read by rows.

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A365665 Expansion of Sum_{0

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A365666 Expansion of Sum_{0

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