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 1-7 of 7 results.

A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 50, 83, 147, 249, 435, 742, 1288, 2207, 3819, 6561, 11333, 19497, 33640, 57915, 99874, 172020, 296550, 510886, 880580, 1517226, 2614889, 4505745, 7765094, 13380640, 23059193, 39735969, 68476885, 118001888
Offset: 0

Views

Author

Paul D. Hanna, Aug 10 2002

Keywords

Comments

Recurrence relation involves the Moebius function.
Radius of convergence of A(x) is r=0.5802946238073267...
Related limits are
lim_{n->infinity} a(n) r^n = 0.6303632342... and
lim_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
From Gary W. Adamson, Aug 11 2016: (Start)
The definition in the heading follows from the INVERTi transform of (1, 2, 3, 6, 9, 17, ...) equals -mu(n) for n >= 2 (cf. A157658).
Then for example, a(6) = 17 = (1, 1, 0, 1, -1, 1) dot (9, 6, 3, 2, 1, 1) = (9 + 6 + 0 + 2 - 1 + 1); in agreement with the first example. (End)

Examples

			a(6) = -mu(2)a(5) - mu(3)a(4) - mu(4)a(3) - mu(5)a(2) - mu(6)a(1) - mu(7)a(0) = 9 + 6 + 0 + 2 - 1 + 1 = 17.
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 17*x^6 + 28*x^7 + 50*x^8 + 83*x^9 + 147*x^10 + 249*x^11 + 435*x^12 + ...
where
1/A(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 - x^18 + x^20 + x^21 - x^22 + x^25 - x^28 - x^29 - x^30 + ... + mu(n)*x^n +...
Also, g.f. A(x) satisfies:
x*A(x) = x*A(x)/A(x*A(x)) + x^2*A(x)^2/A(x^2*A(x)^2) + x^3*A(x)^3/A(x^3*A(x)^3) + x^4*A(x)^4/A(x^4*A(x)^4) + x^5*A(x)^5/A(x^5*A(x)^5) + ...

Links

Crossrefs

Cf. A073777, A068341, A070965, A008683, A185694, A157658.

Programs

  • Haskell
    a073776 n = a073776_list !! (n-1)
a073776_list = 1 : f [1] where
   f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms
   ms = map negate $ tail a008683_list
-- Reinhard Zumkeller, Nov 03 2015
  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[-MoebiusMu[k + 1]*a[n - k], {k, 1, n}]; Array[a,35,0] (* Jean-François Alcover, Apr 11 2011 *)
  • PARI
    {a(n) = my(A=[1,1],F); for(i=1,n, A=concat(A,0); F=Ser(A); A = Vec(sum(m=1,#A, subst(x/F, x, x^m*F^m))) ); A[n+1]}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 19 2016

Formula

G.f.: A(x) = x / (Sum_{n>=1} mu(n)*x^n), A(0)=1, where mu(n) = Moebius function of n.
G.f. A(x) satisfies: x*A(x) = Sum_{n>=1} x^n*A(x)^n / A( x^n*A(x)^n ). - Paul D. Hanna, Apr 19 2016

A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -2, -3, -2, 3, 8, 8, -2, -16, -24, -10, 24, 59, 54, -11, -117, -174, -90, 162, 431, 449, -20, -835, -1393, -848, 1062, 3352, 3748, 317, -6257, -11134, -7583, 7294, 25956, 30786, 5217, -46545, -88132, -65062, 48534, 199234, 249263, 63034, -342174, -691679, -554002
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

Comments

Invert transform of A008683.

Links

Crossrefs

Cf. A008683, A050385, A068341, A073776, A073777, A117209, A117210, A185694, A280194.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      numtheory[mobius](j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    nmax = 47; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[MoebiusMu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]
  • PARI
    my(N=66, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, moebius(k)*x^k))) \\ Seiichi Manyama, Apr 06 2022
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*a(n-k))); \\ Seiichi Manyama, Apr 06 2022

Formula

G.f.: 1/(1 - Sum_{k>=1} A008683(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * a(n-k). - Seiichi Manyama, Apr 06 2022

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}]

A152904 Triangle read by rows: T(n,k) = A008683(n-k+1); 1<=k<=n; mu(n) "decrescendo".

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 0, -1, -1, 1, -1, 0, -1, -1, 1, 1, -1, 0, -1, -1, 1, -1, 1, -1, 0, -1, -1, 1, 0, -1, 1, -1, 0, -1, -1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1
Offset: 1

Views

Author

Gary W. Adamson, Dec 14 2008

Keywords

Examples

			Triangle begins
   1;
  -1, 1;
  -1, -1, 1;
   0, -1, -1, 1;
  -1, 0, -1, -1, 1;
   1, -1, 0, -1, -1, 1;
  -1, 1, -1, 0, -1, -1, 1;
   0, -1, 1, -1, 0, -1, -1, 1;
   0, 0, -1, 1, -1, 0, -1, -1, 1;
   ...
Production matrix begins
  -1, 1,
  -2, 0, 1,
  -3, 0, 0, 1,
  -6, 0, 0, 0, 1,
  -9, 0, 0, 0, 0, 1,
  -17, 0, 0, 0, 0, 0, 1,
  -28, 0, 0, 0, 0, 0, 0, 1,
  -50, 0, 0, 0, 0, 0, 0, 0, 1,
  -83, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  -147, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ...
  where first column is -A073776(n+1). - _Paul Barry_, Feb 10 2011

Links

  • E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Crossrefs

Cf. A008683, A002321, A073776, A152901, A152902.
Row sums = A002321, the Mertens function. A185694 is an eigensequence.

Formula

Triangle read by rows, T(n,k) = A008683(n-k+1) = A008683 in every column = A008683 "decrescendo"d by rows.

A356907 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, 2, -2, 2, 0, -4, 12, -22, 34, -42, 38, -6, -68, 202, -394, 616, -782, 730, -204, -1104, 3486, -6994, 11142, -14452, 14026, -5296, -17558, 60042, -123860, 201128, -266384, 268176, -124034, -273626, 1030396, -2188864, 3624290, -4898740, 5101306, -2744408
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 02 2022

Keywords

Crossrefs

Cf. A008836, A118205, A118206, A118207, A118208, A185694, A307076, A307240.

Programs

  • Mathematica
    nmax = 40; 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, 40}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A008836(k) * a(n-k).
Showing 1-7 of 7 results.

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