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.

Showing 1-5 of 5 results.

A283119 Expansion of exp( Sum_{n>=1} sigma(6*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, 12, 86, 469, 2141, 8594, 31247, 104945, 330094, 982284, 2786861, 7584060, 19893185, 50494558, 124437410, 298555264, 699017259, 1600364304, 3589048673, 7896510620, 17067607791, 36283650153, 75947406513, 156672628539, 318804641925, 640390347979
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2017

Keywords

Comments

sigma(6*n) = A000203(6*n), the sum of divisors of 6*n (A224613).

Examples

			G.f.: A(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 + ...
log(A(x)) = 12*x + 28*x^2/2 + 39*x^3/3 + 60*x^4/4 + 72*x^5/5 + 91*x^6/6 + 96*x^7/7 + 124*x^8/8 + ... + sigma(6*n)*x^n/n + ...

Links

Crossrefs

Cf. A224613 (sigma(6*n)), A283164 (exp( Sum_{n>=1} -sigma(6*n)*x^n/n )).
Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), this sequence (k=6), A283077 (k=7), A283120 (k=8), A283121 (k=9).

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - x^(2 i))^4*(1 - x^(3 i))^3/((1 - x^i)^12*(1 - x^(6 i))), {i, n}], {x, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Mar 01 2017 *)

Formula

G.f.: Product_{n>=1} (1 - x^(2*n))^4 * (1 - x^(3*n))^3/((1 - x^n)^12 * (1 - x^(6*n))).
a(n) = (1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 55^(7/4) * exp(sqrt(55*n)*Pi/3) / (41472*sqrt(3)*n^(9/4)). - Vaclav Kotesovec, Mar 20 2017

A283121 Expansion of exp( Sum_{n>=1} sigma(9*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, 13, 104, 633, 3224, 14404, 58151, 216294, 751582, 2464860, 7689669, 22961822, 65955677, 182985947, 492016590, 1285829996, 3274100475, 8139933477, 19795490575, 47165634583, 110259083454, 253208634687, 571880965638, 1271549402110, 2785836824325, 6019078365425
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2017

Keywords

Examples

			G.f.: A(x) = 1 + 13*x + 104*x^2 + 633*x^3 + 3224*x^4 + 14404*x^5 + ...
log(A(x)) = 13*x + 39*x^2/2 + 40*x^3/3 + 91*x^4/4 + 78*x^5/5 + 120*x^6/6 + 104*x^7/7 + 195*x^8/8 + ... + sigma(9*n)*x^n/n + ...

Links

Crossrefs

Cf. A283123 (sigma(9*n)), A283169 (exp( Sum_{n>=1} -sigma(9*n)*x^n/n )).
Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), A283077 (k=7), A283120 (k=8), this sequence (k=9).

Formula

G.f.: Product_{n>=1} (1 - x^(3*n))^4/(1 - x^n)^13.
a(n) = (1/n)*Sum_{k=1..n} sigma(9*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 1225 * sqrt(35) * exp(sqrt(70*n)*Pi/3) / (559872*n^3). - Vaclav Kotesovec, Mar 20 2017

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22527, 62329, 164516, 416948, 1019690, 2416246, 5565864, 12498215, 27421815, 58903768, 124088548, 256749822, 522450250, 1046735092, 2066948472, 4026431543, 7743987036, 14715788745, 27648250012, 51390298666, 94550761844
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2017

Keywords

Examples

			G.f.: A(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + ...
log(A(x)) = 8*x + 24*x^2/2 + 32*x^3/3 + 56*x^4/4 + 48*x^5/5 + 96*x^6/6 + 57*x^7/7 + 120*x^8/8 + ... + sigma(7*n)*x^n/n + ...

Links

Crossrefs

Cf. A282942 (Product_{n>=1} (1 - x^n)^8/(1 - x^(7*n))), A283078 (sigma(7*n)).
Cf. exp( Sum_{n>=1} sigma(k*n)*x^n/n ): A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), this sequence (k=7), A283120 (k=8), A283121 (k=9).

Formula

G.f.: exp( Sum_{n>=1} sigma(7*n)*x^n/n ).
a(n) = (1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 3025 * exp(sqrt(110*n/21)*Pi) / (28224*sqrt(14)*n^(5/2)). - Vaclav Kotesovec, Mar 20 2017

A283168 Expansion of exp( Sum_{n>=1} -sigma(8*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -15, 97, -350, 770, -1133, 1540, -2731, 4230, -3960, 3402, -6580, 9167, -5390, 4310, -11061, 12320, -5306, 2030, -7530, 14784, -4340, -10119, -9240, 20090, 11438, -17275, -4928, 2270, 14080, -26840, 7700, 16646, 24640, -53760, 7449, 10780, 46200, -61600
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2017

Keywords

Links

Crossrefs

Cf. A283120 (exp( Sum_{n>=1} sigma(8*n)*x^n/n )), A283122 (sigma(8*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), this sequence (k=8), A283169 (k=9).

Formula

G.f.: Product_{n>=1} (1 - x^n)^15/(1 - x^(2*n))^7.
a(n) = -(1/n)*Sum_{k=1..n} sigma(8*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A319362 a(n) = [x^n] exp(Sum_{k>=1} sigma(n*k)*x^k/k).

Original entry on oeis.org

1, 1, 8, 39, 385, 917, 31247, 22527, 1081986, 2464860, 50099635, 14931071, 19684696065, 394805109, 82267017929, 496514888157, 11386442827781, 284625019799, 3469798073972537, 7725084195239, 136470024990370842, 28400489198168457, 241211623942678951, 5776331152550399
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

Links

Crossrefs

Cf. A000041, A182818, A182819, A182820, A182821, A283077, A283119, A283120, A283121, A319363.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[DivisorSigma[1, n k] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Showing 1-5 of 5 results.

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