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.

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A302831 Expansion of (1/(1 - x))*Product_{k>=1} (1 + k*x^k).

Original entry on oeis.org

1, 2, 4, 9, 16, 31, 56, 99, 163, 283, 469, 757, 1220, 1915, 3020, 4748, 7273, 11014, 16789, 25033, 37480, 55782, 82206, 120033, 174762, 253092, 364276, 523814, 749438, 1064853, 1509561, 2128227, 2986392, 4186093, 5832169, 8121130, 11272081, 15576076, 21446615, 29479186, 40360980
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

Comments

Partial sums of A022629.

Links

Crossrefs

Cf. A000009, A022629, A036469, A302830, A302832.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, i*b(n-i, i-1))))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2018
  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*exp(Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j).

A303902 Expansion of (1 - x^2)*Product_{k>=2} (1 + x^k)^k.

Original entry on oeis.org

1, 0, 1, 3, 3, 8, 12, 21, 34, 59, 93, 150, 242, 377, 595, 922, 1419, 2171, 3310, 4988, 7507, 11218, 16674, 24676, 36353, 53295, 77828, 113209, 163989, 236736, 340517, 488108, 697407, 993350, 1410455, 1996968, 2819280, 3969260, 5573541, 7806141, 10905640, 15199138, 21133212
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

First differences of A026007.

Links

Crossrefs

Cf. A002865, A026007, A087897, A191659, A298598, A302832.

Programs

  • Mathematica
    nmax = 42; CoefficientList[Series[(1 - x^2) Product[(1 + x^k)^k, {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[(1 - x) Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: (1 - x)*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * Zeta(3)^(1/2) / (2^(13/12) * sqrt(Pi) * n). - Vaclav Kotesovec, May 04 2018
Showing 1-2 of 2 results.

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