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.

Previous Showing 11-15 of 15 results.

A255837 G.f.: Product_{k>=1} (1+x^k)^(3*k+2).

Original entry on oeis.org

1, 5, 18, 61, 182, 506, 1338, 3369, 8172, 19197, 43833, 97636, 212748, 454461, 953505, 1968095, 4001627, 8024295, 15885484, 31074351, 60111277, 115071431, 218126868, 409662895, 762679151, 1408172844, 2579599582, 4690277001, 8467363674, 15182486586
Offset: 0

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Author

Vaclav Kotesovec, Mar 07 2015

Keywords

Comments

In general, if g.f. = Product_{k>=1} (1+x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(1/6) * exp(-c^2 * Pi^4 / (1296*m*Zeta(3)) + (c * Pi^2 * n^(1/3)) / (2^(5/3) * 3^(4/3) * (m*Zeta(3))^(1/3)) + 3^(4/3) * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(4/3)) / (2^(m/12 + c/2 + 2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Mar 08 2015

Links

Crossrefs

Cf. A026007 (k), A219555 (k+1), A052812 (k-1), A255834 (2*k+1), A255835 (2*k-1), A255836 (3*k+1).
Cf. A255803.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+x^k)^(3*k+2),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4/(972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(2/3) * 3^(5/3) * Zeta(3)^(1/3)) + 3^(5/3)/2^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(23/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

A261384 Expansion of Product_{k>=1} (1+x^k)^(2*k-1) / (1-x^k)^(2*k).

Original entry on oeis.org

1, 3, 12, 39, 117, 331, 893, 2307, 5766, 13986, 33046, 76302, 172567, 383013, 835731, 1795236, 3801105, 7941439, 16386777, 33423342, 67435311, 134675784, 266385932, 522135379, 1014643823, 1955656848, 3740191268, 7100290646, 13383997996, 25058666367
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 17 2015

Keywords

Comments

Convolution of A161870 and A255835.

Links

Crossrefs

Cf. A006950, A015128, A156616, A161870, A255835, A261386.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1+x^k)^(2*k-1)/(1-x^k)^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (7*Zeta(3))^(2/9) * exp(1/6 - Pi^4/(6048*Zeta(3)) - Pi^2 * n^(1/3) / (12*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * 2^(1/6) * sqrt(3*Pi) * n^(13/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A305655 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 3, 13, 54, 290, 1674, 10857, 76398, 580230, 4706734, 40598349, 370694845, 3569027696, 36100349833, 382360758863, 4228730647420, 48716663849192, 583403253712747, 7248883337962522, 93291181556742684, 1241632098163126324, 17064777292709034968, 241874821482784132204
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2018

Keywords

Links

Crossrefs

Cf. A027998, A255835, A281156, A293554, A305206, A305654.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 + x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

Formula

a(n) = [x^n] Product_{k>=1} (1 + x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

A363599 Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.

Original entry on oeis.org

1, 0, 3, 8, 18, 48, 109, 264, 594, 1360, 2988, 6552, 14115, 30048, 63288, 131800, 271953, 555792, 1126583, 2264472, 4518051, 8948544, 17603781, 34405272, 66828247, 129040704, 247765665, 473160696, 898924929, 1699331808, 3197083220, 5987288352, 11162934948
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2023

Keywords

Crossrefs

Cf. A027998, A052812, A255835, A363600.
Cf. A363601.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2-1)))

Formula

G.f.: Product_{k>=1} (1+x^k)^(k^2-1).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2-1) ) * a(n-k).

Extensions

Name suggested by Joerg Arndt, Jun 11 2023

A319669 Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

Original entry on oeis.org

1, -1, -3, -2, 1, 10, 13, 15, -1, -30, -63, -89, -80, -14, 131, 304, 493, 561, 434, -32, -836, -1895, -2960, -3583, -3240, -1338, 2401, 8004, 14499, 20494, 23369, 20401, 8567, -13741, -46408, -85717, -124027, -149612, -147167, -101002, 2520, 168026, 388077, 634914
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2018

Keywords

Crossrefs

Cf. A000203, A001157, A073592, A253289, A255835, A276551, A278945, A285069.

Programs

  • Maple
    a:=series(mul((1-x^k)^(2*k-1),k=1..100),x=0,44): seq(coeff(a,x,n),n=0..43); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 43; CoefficientList[Series[Product[(1 - x^k)^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 2 DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - 2 d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: exp(Sum_{k>=1} (sigma_1(k) - 2*sigma_2(k))*x^k/k).
Previous Showing 11-15 of 15 results.

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