A283168 -id:A283168 - cp's OEIS frontend

A283120 Expansion of exp( Sum_{n>=1} sigma(8n)x^n/n ) in powers of x.

1, 15, 128, 815, 4289, 19663, 81057, 306799, 1081986, 3594142, 11338690, 34193246, 99080387, 277046893, 750192227, 1973050940, 5053026949, 12628736331, 30859262181, 73849589786, 173333118663, 399528823032, 905418038792, 2019454523623, 4437187104779

Offset: 0

Seiichi Manyama, Mar 01 2017

nonn

			G.f.: A(x) = 1 + 15*x + 128*x^2 + 815*x^3 + 4289*x^4 + 19663*x^5 + ...
log(A(x)) = 15*x + 31*x^2/2 + 60*x^3/3 + 63*x^4/4 + 90*x^5/5 + 124*x^6/6 + 120*x^7/7 + 127*x^8/8 + ... + sigma(8*n)*x^n/n + ...

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

Cf. A283122 (sigma(8*n)), A283168 (exp( Sum_{n>=1} -sigma(8*n)*x^n/n )).

Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), A283077 (k=7), this sequence (k=8), A283121 (k=9).

G.f.: Product_{n>=1} (1 - x^(2*n))^7/(1 - x^n)^15.
a(n) = (1/n)*Sum_{k=1..n} sigma(8*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 529 * 23^(1/4) * exp(sqrt(23*n/3)*Pi) / (73728 * 3^(1/4) * n^(11/4)). - Vaclav Kotesovec, Mar 20 2017

A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -12, 58, -133, 95, 194, -418, 97, 325, -99, -238, 169, -217, 131, 190, -145, 441, -647, 169, -527, 72, 1129, 313, -972, 2, -491, -565, 1944, -1175, -216, 972, 863, -1259, 288, 0, -1155, -1355, -207, 2925, 1753, 1402, -2387, -2257, -1030, 315, 432, -72, 1621, 358

Offset: 0

Seiichi Manyama, Mar 02 2017

sign

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

Cf. A224613 (sigma(6*n)), A283119 (exp( Sum_{n>=1} sigma(6*n)*x^n/n )).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), this sequence (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

G.f.: Product_{n>=1} (1 - x^n)^12 * (1 - x^(6*n))/((1 - x^(2*n))^4 * (1 - x^(3*n))^3).

a(n) = -(1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283169 Expansion of exp( Sum_{n>=1} -sigma(9n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -13, 65, -126, -117, 988, -1377, -1157, 5382, -4419, -4212, 12519, -11179, -5058, 27378, -23005, -16488, 44343, -30249, -18513, 73710, -56259, -38741, 93483, -69570, -23778, 137266, -90396, -74079, 140292, -108621, -39249, 222624, -145710, -99234

Offset: 0

Seiichi Manyama, Mar 02 2017

sign

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

Cf. A283121 (exp( Sum_{n>=1} sigma(9*n)*x^n/n )), A283123 (sigma(9*n)).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), A283168 (k=8), this sequence (k=9).

G.f.: Product_{n>=1} (1 - x^n)^13/(1 - x^(3*n))^4.

a(n) = -(1/n)*Sum_{k=1..n} sigma(9*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283163 Expansion of exp( Sum_{n>=1} -sigma(4n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -7, 17, -14, 2, -21, 36, 13, -26, -24, 10, 12, -17, 34, 22, 19, -96, -10, 14, 38, 0, 12, -23, 72, -38, -2, -11, -64, -34, 0, 72, 84, -26, 0, 0, -79, 60, 24, -32, -58, -7, -84, 50, 26, 120, 0, 0, 46, -34, -64, 10, -119, 70, 0, 22, -70, 36, 37, -120, 0

Offset: 0

Seiichi Manyama, Mar 02 2017

sign

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

Cf. A182820 (exp( Sum_{n>=1} sigma(4*n)*x^n/n )), A193553 (sigma(4*n)).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), this sequence (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

G.f.: Product_{n>=1} (1 - x^n)^7/(1 - x^(2*n))^3.

a(n) = -(1/n)*Sum_{k=1..n} sigma(4*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

cp's OEIS Frontend

A283120 Expansion of exp( Sum_{n>=1} sigma(8n)x^n/n ) in powers of x.

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A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.

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A283169 Expansion of exp( Sum_{n>=1} -sigma(9n)x^n/n ) in powers of x.

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A283163 Expansion of exp( Sum_{n>=1} -sigma(4n)x^n/n ) in powers of x.

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cp's OEIS Frontend

A283120 Expansion of exp( Sum_{n>=1} sigma(8*n)*x^n/n ) in powers of x.

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Author

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Formula

A283164 Expansion of exp( Sum_{n>=1} -sigma(6*n)*x^n/n ) in powers of x.

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Formula

A283169 Expansion of exp( Sum_{n>=1} -sigma(9*n)*x^n/n ) in powers of x.

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Keywords

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Crossrefs

Formula

A283163 Expansion of exp( Sum_{n>=1} -sigma(4*n)*x^n/n ) in powers of x.

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Formula

A283120 Expansion of exp( Sum_{n>=1} sigma(8n)x^n/n ) in powers of x.

A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.

A283169 Expansion of exp( Sum_{n>=1} -sigma(9n)x^n/n ) in powers of x.

A283163 Expansion of exp( Sum_{n>=1} -sigma(4n)x^n/n ) in powers of x.