A283224 -id:A283224 - cp's OEIS frontend

A283238 Expansion of exp( Sum_{n>=1} sigma_2(3n)x^n/n ) in powers of x.

1, 10, 75, 447, 2335, 10977, 47667, 193775, 745819, 2738568, 9653342, 32821384, 108061167, 345622069, 1076767956, 3275039352, 9743874779, 28405248949, 81256123017, 228383836091, 631427595230, 1718990010867, 4612234492354, 12206598044861, 31889379237288

Offset: 0

Seiichi Manyama, Mar 03 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. exp( Sum_{n>=1} sigma_k(3*n)*x^n/n ): A182819 (k=1), this sequence (k=2).

Cf. exp( Sum_{n>=1} sigma_2(m*n)*x^n/n ): A000219 (m=1), A283224 (m=2), this sequence (m=3).

a(n) = (1/n)*Sum_{k=1..n} sigma_2(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A099978 Bisection of A001157: a(n) = sigma_2(2n-1).

Original entry on oeis.org

1, 10, 26, 50, 91, 122, 170, 260, 290, 362, 500, 530, 651, 820, 842, 962, 1220, 1300, 1370, 1700, 1682, 1850, 2366, 2210, 2451, 2900, 2810, 3172, 3620, 3482, 3722, 4550, 4420, 4490, 5300, 5042, 5330, 6510, 6100, 6242, 7381, 6890, 7540, 8420, 7922, 8500

Offset: 1

N. J. A. Sloane, Nov 19 2004

			From _M. F. Hasler_, Mar 06 2017: (Start)
a(1) = sigma_2(2*1-1) = 1.
a(2) = sigma_2(2*2-1) = 1 + 3^2 = 10.
a(5) = sigma_2(2*5-1) = 1 + 3^2 + 9^2 = 91. (End)
G.f.: A(x) = x + 10*x^2 + 26*x^3 + 50*x^4 + 91*x^5 + 122*x^6 + 170*x^7 + 260*x^8 + 290*x^9 + 362*x^10 + 500*x^11 + 530*x^12 + ... where A(x) = x/(1 - x) + 3^2*x^2/(1 - x^3) + 5^2*x^3/(1 - x^5) + 7^2*x^4/(1 - x^7) + 9^2*x^5/(1 - x^9) + .... - _Paul D. Hanna_, Jun 23 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A099979(n) = sigma_2(2n), the other bisection of A001157.

Cf. A283224.

with(numtheory): A099978 := n->sigma[2](2*n-1): seq(A099978(n), n=1..60);

DivisorSigma[2, Range[1, 91, 2]] (* Amiram Eldar, Aug 17 2019 *)

A099978(n)=sigma(2*n-1,2) \\ M. F. Hasler, Mar 06 2017

a(n) = A001157(2n-1) = sigma_2(2n-1). - M. F. Hasler, Mar 06 2017
Sum_{k=1..n} a(k) ~ 7*zeta(3)*n^3/6. - Vaclav Kotesovec, Aug 07 2022
G.f.: Sum_{n>=1} (2*n-1)^2 * x^n / (1 - x^(2*n-1)). - Paul D. Hanna, Jun 23 2025

More terms from Emeric Deutsch, Dec 07 2004

Edited by M. F. Hasler, Mar 06 2017

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 9, 77, 534, 3320, 18933, 100770, 506697, 2428161, 11161765, 49469005, 212246744, 884491121, 3589900607, 14223638534, 55122970206, 209307080221, 779837798559, 2854660220661, 10278494869342, 36439277959593, 127311828611819, 438712861233581

Offset: 0

Seiichi Manyama, Mar 03 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. exp( Sum_{n>=1} sigma_k(2*n)*x^n/n ): A182818 (k=1), A283224 (k=2), this sequence (k=3).

Cf. exp( Sum_{n>=1} sigma_3(m*n)*x^n/n ): A023871 (m=1), this sequence (m=2), A283244 (m=3).

a(n) = (1/n)*Sum_{k=1..n} sigma_3(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -5, 2, 15, 12, -36, -92, -17, 167, 358, 283, -293, -1321, -2012, -1101, 2299, 7296, 10505, 6901, -7705, -31240, -52490, -51336, -6032, 91521, 217064, 303776, 250595, -36282, -575622, -1234465, -1684515, -1448538, -66980, 2610835, 6087681, 8990575

Offset: 0

Seiichi Manyama, Mar 03 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A283224 (exp( Sum_{n>=1} sigma_2(2*n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(2*n)*x^n/n ): A115110 (k=1), this sequence (k=2).
Cf. exp( Sum_{n>=1} -sigma_2(m*n)*x^n/n ): A073592 (m=1), this sequence (m=2), A283243 (m=3).

a(n) = -(1/n)*Sum_{k=1..n} sigma_2(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

cp's OEIS Frontend

A283238 Expansion of exp( Sum_{n>=1} sigma_2(3n)x^n/n ) in powers of x.

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A099978 Bisection of A001157: a(n) = sigma_2(2n-1).

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A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

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A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.

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cp's OEIS Frontend

A283238 Expansion of exp( Sum_{n>=1} sigma_2(3*n)*x^n/n ) in powers of x.

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Author

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Formula

A099978 Bisection of A001157: a(n) = sigma_2(2n-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2*n)*x^n/n ) in powers of x.

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Author

Keywords

Links

Crossrefs

Formula

A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2*n)*x^n/n ) in powers of x.

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Formula

A283238 Expansion of exp( Sum_{n>=1} sigma_2(3n)x^n/n ) in powers of x.

A282327 Expansion of exp( Sum_{n>=1} sigma_3(2n)x^n/n ) in powers of x.

A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.