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 1-4 of 4 results.

A349450 Dirichlet inverse of right-shifted Catalan numbers [as when started from A000108(0): 1, 1, 2, 5, 14, 42, etc.].

Original entry on oeis.org

1, -1, -2, -4, -14, -38, -132, -420, -1426, -4834, -16796, -58688, -208012, -742636, -2674384, -9693976, -35357670, -129641774, -477638700, -1767253368, -6564119892, -24466233428, -91482563640, -343059494120, -1289904147128, -4861945985428, -18367353066440, -69533549429280, -263747951750360, -1002242211282032
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A000108, A035010, A035102.
Cf. also A349449, A349451, A349452, A349453, A349564.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * CatalanNumber[n/# - 1] &, # < n &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A000108(n) = binomial(2*n, n)/(n+1);
memoA349450 = Map();
A349450(n) = if(1==n,1,my(v); if(mapisdefined(memoA349450,n,&v), v, v = -sumdiv(n,d,if(dA000108((n/d)-1)*A349450(d),0)); mapput(memoA349450,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A000108((n/d)-1) * a(d).
For n > 1, a(n) = -A035010(n) = A035102(n) - A000108(n-1).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Catalan(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022
x = Sum_{n>=1} a(n) * C(x^n) where C(x) = (1 - sqrt(1-4*x))/2 is the g.f. of the Catalan numbers (A000108). - Paul D. Hanna, Nov 27 2024

A349563 Dirichlet convolution of right-shifted Catalan numbers with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

Original entry on oeis.org

1, -1, -2, -1, -2, 18, 68, 311, 1182, 4370, 15772, 56754, 203916, 734636, 2658096, 9661591, 35292134, 129511602, 477376556, 1766730706, 6563071700, 24464139348, 91478369336, 343051112482, 1289887370140, 4861912443284, 18367285959072, 69533415236716, 263747683314904, 1002241674463968, 3814985428350480, 14544633872450487
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Comments

Dirichlet convolution with A034729 gives A034731.

Links

Crossrefs

Cf. A000108, A011782, A349452, A349564 (Dirichlet inverse).
Cf. also A034729, A034731, A349565, A349567, A349569.

Programs

  • Mathematica
    s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, CatalanNumber[# - 1] * s[n/#] &]; Array[a, 32] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A000108(n) = (binomial(2*n, n)/(n+1));
A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349563(n) = sumdiv(n,d,A000108(d-1)*A349452(n/d));

Formula

a(n) = Sum_{d|n} A000108(d-1) * A349452(n/d).

A349568 Dirichlet convolution of A011782 [2^(n-1)] with A349453 (Dirichlet inverse of A133494, 3^(n-1)).

Original entry on oeis.org

1, -1, -5, -16, -65, -187, -665, -1984, -6260, -18895, -58025, -174016, -527345, -1583407, -4765595, -14307568, -42981185, -128980852, -387158345, -1161657760, -3485726195, -10458022927, -31376865305, -94134053296, -282412754000, -847252941535, -2541798630320, -7625456893096, -22876524019505, -68629821114805
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Comments

Dirichlet convolution of this sequence with A034754 produces A034738.

Links

Crossrefs

Cf. A011782, A133494, A349453, A349567 (Dirichlet inverse).
Cf. also A034738, A034754, A349564, A349566, A349570.

Programs

  • Mathematica
    s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 3^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, 2^(# - 1) * s[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A133494(n) = max(1, 3^(n-1));
memoA349453 = Map();
A349453(n) = if(1==n,1,my(v); if(mapisdefined(memoA349453,n,&v), v, v = -sumdiv(n,d,if(dA133494(n/d)*A349453(d),0)); mapput(memoA349453,n,v); (v)));
A349568(n) = sumdiv(n,d,(2^(d-1)) * A349453(n/d));

Formula

a(n) = Sum_{d|n} 2^(d-1) * A349453(n/d).

A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 1, 4, 11, 24, 57, 112, 244, 480, 1013, 1972, 4083, 8064, 16331, 32512, 65519, 130488, 262125, 523244, 1048377, 2095104, 4194281, 8384176, 16777136, 33546240, 67108096, 134201316, 268435427, 536836584, 1073741793, 2147418112, 4294964213, 8589803520, 17179868787, 34359470272, 68719476699, 137438429184, 274877894643
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Comments

Dirichlet convolution of this sequence with phi (A000010) is A000740, with sigma (A000203) it is A034729, and with A018804 it is A034738.

Links

Crossrefs

Cf. A011782, A055615, A349569 (Dirichlet inverse).
Cf. also A000010, A000740, A000203, A018804, A034729, A034738, A349564, A349566, A349568.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # * MoebiusMu[#] * 2^(n/# - 1) &]; Array[a, 40] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A055615(n) = (n*moebius(n));
A349570(n) = sumdiv(n,d,(2^(d-1)) * A055615(n/d));

Formula

a(n) = Sum_{d|n} 2^(d-1) * A055615(n/d).
Showing 1-4 of 4 results.

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