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 1-3 of 3 results.

A321384 a(1) = 1; a(n+1) = -Sum_{d|n} d*a(d).

Original entry on oeis.org

1, -1, 1, -4, 17, -86, 514, -3599, 28809, -259285, 2592766, -28520427, 342245654, -4449193503, 62288705445, -934330581764, 14949289337033, -254137918729562, 4574482536873349, -86915168200593632, 1738303364014465422, -36504370644303777464, 803096154174654583783, -18471211546017055427010
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 08 2018

Keywords

Links

Crossrefs

Cf. A281487, A318372.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -Sum[d a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 24}]
  • PARI
    a(n) = if (n==1, 1, -sumdiv(n-1, d, d*a(d))); \\ Michel Marcus, Nov 09 2018

Formula

L.g.f.: log(Product_{n>=1} (1 - x^n)^a(n)) = Sum_{n>=1} a(n+1)*x^n/n.
a(n) ~ -(-1)^n * c * (n-1)!, where c = 0.7144978951771230847588633755835851845867260778566988217176856019246992... - Vaclav Kotesovec, Nov 09 2018

A307992 G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(x) + 2*A(x^2) + 3*A(x^3) + ...).

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 9, 20, 16, 38, 28, 61, 39, 110, 52, 149, 84, 225, 101, 317, 120, 454, 175, 543, 198, 823, 243, 940, 327, 1259, 356, 1601, 387, 2051, 515, 2270, 623, 3114, 660, 3373, 829, 4381, 870, 5145, 913, 6264, 1245, 6683, 1292, 8776, 1404, 9477, 1724
Offset: 1

Views

Author

Ilya Gutkovskiy, May 09 2019

Keywords

Links

Crossrefs

Cf. A007439, A038046, A144366, A318372.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, signum(n), (m->
      m*add(a(d)/d, d=numtheory[divisors](m)))(n-2))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 09 2019
  • Mathematica
    terms = 57; A[] = 0; Do[A[x] = x + x^2 (1 + Sum[k A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x + x^2 (1 + Sum[a[k] x^k/(1 - x^k)^2, {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 57}]
a[n_] := a[n] = Sum[d a[(n - 2)/d], {d, Divisors[n - 2]}]; a[1] = a[2] = 1; Table[a[n], {n, 1, 57}]

Formula

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

A318600 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+1)*d*a(d).

Original entry on oeis.org

1, 1, 1, 4, 13, 66, 394, 2759, 22053, 198481, 1984746, 21832207, 261986098, 3405819275, 47681467093, 715222006464, 11443552081333, 194540385382662, 3501726936689833, 66532811797106828, 1330656235940151698, 27943780954743188420, 614763181004328313035, 14139553163099551199806
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Links

Crossrefs

Cf. A004111, A157311, A318372.

Programs

  • Maple
    f:= proc(n) option remember;
add((-1)^((n-1)/d+1)*d*procname(d), d = numtheory:-divisors(n-1))
end proc:
f(1):= 1:
map(f, [$1..30]); # Robert Israel, Aug 30 2018
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d + 1) d a[d] , {d, Divisors[n - 1]}]; Table[a[n], {n, 24}]

Formula

L.g.f.: log(Product_{n>=1} (1 + x^n)^a(n)) = Sum_{n>=1} a(n+1)*x^n/n.
Showing 1-3 of 3 results.

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