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 1-6 of 6 results.

A308367 Expansion of Sum_{k>=1} x^k/(1 + k*x^k).

Original entry on oeis.org

1, 0, 2, -2, 2, 1, 2, -12, 11, 11, 2, -49, 2, 57, 108, -200, 2, 40, 2, -391, 780, 1013, 2, -5423, 627, 4083, 6644, -4453, 2, -5043, 2, -49680, 59172, 65519, 18028, -251062, 2, 262125, 531612, -861481, 2, -515723, 2, -1049929, 5180382, 4194281, 2, -27246019, 117651
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Links

Crossrefs

Cf. A076717, A087909, A300786.

Programs

  • Mathematica
    nmax = 49; CoefficientList[Series[Sum[x^k /(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 49; CoefficientList[Series[Log[Product[(1 + k x^k)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
Table[Sum[(-d)^(n/d - 1), {d, Divisors[n]}], {n, 1, 49}]
  • PARI
    a(n) = sumdiv(n, d, (-d)^(n/d-1)); \\ Michel Marcus, Mar 22 2021

Formula

L.g.f.: log(Product_{k>=1} (1 + k*x^k)^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-d)^(n/d-1).
a(n) = 2 if n is odd prime.

A294464 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(1/k).

Original entry on oeis.org

1, 1, 2, 12, 36, 300, 2520, 20160, 75600, 2192400, 30996000, 276091200, 2165486400, 19070251200, 968042275200, 41954552640000, 190974944160000, 230641066656000, 95607669148992000, -2052972258809472000, 22839078791168640000, 5074390517301705600000
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Crossrefs

Cf. A022629, A076717, A168243, A294462.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+k*x^k)^(1/k))))

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018
a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} A076717(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jan 21 2025

A300786 L.g.f.: log(Product_{k>=1} (1 + k*x^k)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 3, 10, 7, 26, 24, 50, -33, 163, 38, 122, -188, 170, 108, 1580, -1793, 290, -273, 362, -1678, 9404, 3248, 530, -49092, 16251, 14862, 66340, 14000, 842, -135556, 962, -429057, 547172, 258386, 509500, -1392821, 1370, 1043160, 4813052, -8088838, 1682, -9267612, 1850, 8218844, 53396438
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 12 2018

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 7*x^4/4 + 26*x^5/5 + 24*x^6/6 + 50*x^7/7 - 33*x^8/8 + 163*x^9/9 + 38*x^10/10 + ...
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 15*x^5 + 25*x^6 + 43*x^7 + 64*x^8 + 120*x^9 + 186*x^10 + ... + A022629(n)*x^n + ...

Crossrefs

Cf. A022629, A076717, A078308.

Programs

  • Mathematica
    nmax = 45; Rest[CoefficientList[Series[Log[Product[(1 + k x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[k^2 x^k/(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-d)^(n/d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 45}]

Formula

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

A308366 Expansion of Sum_{k>=1} (-1)^(k+1)*k*x^k/(1 - k*x^k).

Original entry on oeis.org

1, -1, 4, -7, 6, -4, 8, -39, 37, -16, 12, -94, 14, -92, 384, -591, 18, 65, 20, -1542, 2552, -1948, 24, -3606, 3151, -8048, 20440, -30590, 30, 33326, 32, -135455, 178512, -130816, 94968, -35029, 38, -523964, 1596560, -1749734, 42, 2521186, 44, -8374494, 16364502
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A002129, A055225, A076717.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[(-1)^(k + 1) k x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 45; CoefficientList[Series[-Log[Product[(1 - k x^k)^((-1)^(k + 1)/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
Table[Sum[(-1)^(d + 1) d^(n/d), {d, Divisors[n]}], {n, 1, 45}]

Formula

L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^((-1)^(k+1)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(d+1)*d^(n/d).
a(n) = n + 1 if n is odd prime.

A359733 a(n) = (1/2) * Sum_{d|n} (2*d)^(n/d).

Original entry on oeis.org

1, 4, 7, 20, 21, 88, 71, 296, 373, 1084, 1035, 5084, 4109, 16496, 20787, 67728, 65553, 286516, 262163, 1070180, 1189937, 4194568, 4194327, 17760824, 16827241, 67109228, 72150655, 269503660, 268435485, 1104603808, 1073741855, 4303389216, 4476371181
Offset: 1

Views

Author

Seiichi Manyama, Jan 12 2023

Keywords

Crossrefs

Cf. A055225, A076717.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (2*#)^(n/#) &] / 2; Array[a, 33] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (2*d)^(n/d))/2;
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-2*k*x^k)))

Formula

G.f.: Sum_{k>0} k * x^k / (1 - 2 * k* x^k).

A294465 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(-1/k).

Original entry on oeis.org

1, -1, 0, -6, 36, -180, 720, -7560, 236880, -3099600, 15120000, -194594400, 9989179200, -131935003200, 337815878400, -50154760656000, 2018231927712000, -27611162875296000, 363290246871552000, -12648028196067264000, 521752941995725440000
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Crossrefs

Cf. A076717, A294356, A294462, A294463, A294464.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1+k*x^k)^(1/k))))

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018
a(0) = 1 and a(n) = -(n-1)! * Sum_{k=1..n} A076717(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jan 21 2025
Showing 1-6 of 6 results.

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