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 1-9 of 9 results.

A185694 Eigensequence for the Moebius mu triangle A152904.

Original entry on oeis.org

1, -1, 2, -2, 3, -2, 1, 4, -10, 22, -36, 56, -72, 84, -71, 22, 99, -311, 654, -1122, 1702, -2267, 2611, -2312, 795, 2789, -9352, 19802, -34442, 52436, -70521, 81869, -74248, 29099, 79012, -279732, 600894, -1054809, 1616669, -2190772, 2567994
Offset: 0

Views

Author

Paul Barry, Feb 10 2011

Keywords

Comments

First column of the inverse of the number triangle T(n,k)=if(kA152904(n-1,k), if(k=n,1,0)).

Links

Crossrefs

Cf. A008683, A073776, A300663.

Programs

  • Haskell
    a185694 n = a185694_list !! (n-1)
a185694_list = 1 : f [1] where
   f xs = y : f (y : xs) where
          y = sum $ zipWith (*) xs $ map negate a008683_list
-- Reinhard Zumkeller, Nov 03 2015
  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + Sum[MoebiusMu[k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 30 2024 *)

Formula

G.f.: 1 / (1 + Sum_{k>=1} mu(k)*x^k). - Ilya Gutkovskiy, Oct 02 2022

A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

Comments

Exponential transform of A008683.

Examples

			E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...

Links

Crossrefs

Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A307076 Expansion of 1/(1 - Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

Original entry on oeis.org

1, 1, 0, -2, -2, 0, 4, 4, -2, -10, -6, 10, 22, 4, -34, -46, 16, 102, 86, -100, -272, -126, 370, 650, 60, -1138, -1384, 526, 3142, 2532, -2936, -7952, -3440, 10802, 18426, 596, -33344, -38418, 18716, 91934, 68400, -93402, -230962, -86236, 330144, 528880, -17298, -996040
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

Comments

Invert transform of A008836.

Crossrefs

Cf. A008836, A118206, A118207, A300663.

Programs

  • Mathematica
    nmax = 47; CoefficientList[Series[1/(1 - Sum[LiouvilleLambda[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[LiouvilleLambda[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A008836(k)*a(n-k).

A357521 Expansion of Product_{k>=1} (1 - mu(k)*x^k).

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -3, 3, -2, 0, 1, -2, 2, -2, 1, -2, 3, -4, 4, -2, 0, 2, -5, 6, -5, 3, -2, 1, -1, 1, 0, 0, 3, -6, 6, -5, 4, 0, -5, 7, -7, 5, -2, 2, 0, -2, 0, 1, 5, -7, 11, -14, 11, -6, -1, 9, -12, 8, -11, 11, -6, 10, -13, 8, -2, -12, 26, -26, 24, -20, 2, 11, -8, 14, -15, 9
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008683, A117208, A185694, A292561, A300663, A306327, A357524, A357525.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[(1 - MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d MoebiusMu[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A357524 Expansion of Product_{k>=1} 1 / (1 + mu(k)*x^k).

Original entry on oeis.org

1, -1, 2, -1, 2, 0, 1, 2, 0, 3, 0, 4, 1, 4, 2, 4, 4, 4, 5, 6, 6, 6, 8, 8, 10, 9, 11, 12, 13, 14, 17, 17, 20, 19, 23, 24, 28, 27, 30, 34, 34, 40, 41, 47, 48, 50, 56, 62, 64, 71, 72, 80, 85, 91, 99, 104, 113, 112, 128, 135, 147, 153, 159, 176, 180, 196, 210, 220, 233, 240, 264
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008683, A117211, A185694, A300663, A306327, A329069, A357521, A357525.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1/(1 + MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 70}]

A357525 Expansion of Product_{k>=1} (1 + mu(k)*x^k).

Original entry on oeis.org

1, 1, -1, -2, -1, 0, 1, 1, 0, 0, 1, 0, -2, -2, 1, 4, 3, -2, -4, -2, 0, 2, 3, 0, -1, 1, 0, -3, -3, -1, 2, 4, 3, 0, -2, -1, 2, 0, -5, -3, 3, 3, 0, -2, -4, -2, 4, 5, 3, 3, 1, -4, -9, -8, 3, 11, 6, 0, -3, -7, -4, 2, -1, -2, 6, 8, -2, -10, -8, 4, 14, 11, 2, -6, -11, -5
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008683, A087188, A117210, A185694, A300663, A306327, A357521, A357524.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[(1 + MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A352868 Expansion of e.g.f. exp(Sum_{k>=1} mu(k) * x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, -1, -11, -23, -19, 991, 4369, -11311, -356903, 5389471, 7875341, -430708871, -16579950971, 45417621887, 3629980647721, 93982540029601, -1077931879771471, -29167938898699841, -486520057714400603, 7973931691642326281, 205214099791890382621
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2022

Keywords

Examples

			E.g.f.: A(x) = 1 + x - x^2/2! - 11*x^3/3! - 23*x^4/4! - 19*x^5/5! + 991*x^6/6! + 4369*x^7/7! - 11311*x^8/8! - 356903*x^9/9! + 5389471*x^10/10! + ...
where A(x) = exp(x - x^2 - x^3 - x^5 + x^6 - x^7 + ... + mu(n)*x^n +....);
thus, exp(x) = A(x) * A(x^2) * A(x^3) * ... * A(x^n) * ...

Crossrefs

Cf. A008683, A300663, A300673.

Programs

  • Mathematica
    nmax = 20; A[] = 1; Do[A[x] = Exp[x]/Product[A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]*Range[0, nmax]! (* Vaclav Kotesovec, Mar 01 2024 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, moebius(k)*x^k))))
  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*moebius(k)*a(n-k)/(n-k)!));

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * mu(k) * a(n-k)/(n-k)!.
E.g.f. A(x) satisfies Product_{n>=1} A(x^n) = exp(x). - Paul D. Hanna, Feb 29 2024

A352869 Expansion of e.g.f. 1/(1 - Sum_{k>=1} mu(k) * x^k/k!), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 1, -1, -14, -71, -201, 559, 14152, 125772, 568873, -2930247, -100950588, -1263405885, -7645798213, 62733063199, 2644646815760, 42203809509047, 312892097907012, -3774840465405301, -184229592151309092, -3541775382376189109, -30473600413019593651
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2022

Keywords

Crossrefs

Cf. A008683, A300663, A300673.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, moebius(k)*x^k/k!))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n,k) * a(n-k).

A353191 Expansion of e.g.f. 1/(1 - Sum_{k>=1} mu(k) * x^k / k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 1, -2, -22, -134, -418, 1044, 35352, 371256, 2662872, 2256, -348450672, -7383337584, -85166363280, -224652273504, 17983453809024, 500248545941376, 7414246148833152, 13911378371907840, -2620344425592796416, -85017815816225598720, -1321651042532303189760
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2022

Keywords

Crossrefs

Cf. A008683, A300663, A352869, A353189.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * MoebiusMu[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 23, 0] (* Amiram Eldar, Apr 30 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, moebius(k)*x^k/k))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*moebius(j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (k-1)! * mu(k) * binomial(n,k) * a(n-k).
Showing 1-9 of 9 results.

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