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.

A349452 Dirichlet inverse of A011782, 2^(n-1).

Original entry on oeis.org

1, -2, -4, -4, -16, -16, -64, -104, -240, -448, -1024, -1904, -4096, -7936, -16256, -32272, -65536, -129888, -262144, -522176, -1048064, -2093056, -4194304, -8379520, -16776960, -33538048, -67106880, -134184704, -268435456, -536801024, -1073741824, -2147352224, -4294959104, -8589672448, -17179867136, -34359197184
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A011782.
Cf. also A349449, A349450, A349451, A349453 and A349563, A349565, A349567, A349569.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * 2^(n/# - 1) &, # < n &]; Array[a, 36] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    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)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A011782(n/d) * a(d).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} 2^(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022

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

A349451 Dirichlet inverse of Fibonacci numbers, when started from A000045(1): 1, 1, 2, 3, 5, 8, 13, 21, ...

Original entry on oeis.org

1, -1, -2, -2, -5, -4, -13, -16, -30, -45, -89, -122, -233, -351, -590, -944, -1597, -2496, -4181, -6640, -10894, -17533, -28657, -46000, -75000, -120927, -196290, -317018, -514229, -830580, -1346269, -2176288, -3524222, -5699693, -9227335, -14924550, -24157817, -39079807, -63245054, -102320320, -165580141, -267890844
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A000045.
Cf. also A349449, A349450, A349452, A349453, A349566.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * Fibonacci[n/#] &, # < n &]; Array[a, 42] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    memoA349451 = Map();
A349451(n) = if(1==n,1,my(v); if(mapisdefined(memoA349451,n,&v), v, v = -sumdiv(n,d,if(dA349451(d),0)); mapput(memoA349451,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A000045(n/d) * a(d).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Fibonacci(k) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022

A349453 Dirichlet inverse of A133494, 3^(n-1).

Original entry on oeis.org

1, -3, -9, -18, -81, -189, -729, -2052, -6480, -19197, -59049, -175446, -531441, -1589949, -4781511, -14335704, -43046721, -129097152, -387420489, -1162141182, -3486771279, -10459998909, -31381059609, -94142073420, -282429529920, -847285420797, -2541865710960, -7625587899366, -22876792454961, -68630348286531
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A133494.
Cf. also A349449, A349450, A349451, A349452, A349568.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * 3^(n/# - 1) &, # < 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)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A133494(n/d) * a(d).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} 3^(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022
Showing 1-4 of 4 results.

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