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.

Previous Showing 21-29 of 29 results.

A343370 a(1) = 1; a(n) = Sum_{d|n, d < n} (-1)^d * a(d).

Original entry on oeis.org

1, -1, -1, -2, -1, -1, -1, -4, 0, -1, -1, -4, -1, -1, 1, -8, -1, -2, -1, -4, 1, -1, -1, -12, 0, -1, 0, -4, -1, -3, -1, -16, 1, -1, 1, -10, -1, -1, 1, -12, -1, -3, -1, -4, 0, -1, -1, -32, 0, -2, 1, -4, -1, -4, 1, -12, 1, -1, -1, -16, -1, -1, 0, -32, 1, -3, -1, -4, 1, -3, -1, -36, -1, -1, 0, -4, 1, -3, -1, -32, 0
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

Links

Crossrefs

Cf. A008683, A053850 (positions of 0's), A056913 (positions of 1's), A067856, A074206, A307778, A308077, A325144.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1,
      add((-1)^d*a(d), d=numtheory[divisors](n) minus {n}))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 12 2021
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[If[d < n, (-1)^d a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 70}]
  • PARI
    memoA343370 = Map();
A343370(n) = if(1==n,1,my(v); if(mapisdefined(memoA343370,n,&v), v, v = sumdiv(n,d,if(dA343370(d),0)); mapput(memoA343370,n,v); (v))); \\ Antti Karttunen, Jan 02 2023

Formula

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

Extensions

Data section extended up to a(81) by Antti Karttunen, Jan 02 2023

A353972 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsin(x).

Original entry on oeis.org

1, 0, 1, -4, 29, -124, 1583, -17088, 124553, -1152816, 20127867, -262838016, 3978820221, -48595514304, 914656587063, -24441484099584, 370244721585681, -5884988565575424, 162968423791332339, -3855257807841017856, 82014901819948738629, -1934570487417807744000, 58311771938510122952559
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2022

Keywords

Crossrefs

Cf. A001818, A067856, A353818, A353913, A354115, A354171, A354274, A354275, A354276.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = Mod[n, 2] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arcsin(x^k)) / k.

A354274 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsinh(x).

Original entry on oeis.org

1, 0, -1, 4, -11, -4, -547, 7680, -7751, 81744, -3258663, -9474816, -390445563, 233029824, -964154427, 4193551958016, -18431412645519, 71090090006784, -6436900596281679, 17349989459410944, 834261829219880829, -241960391975347200, -1149793471388581053219
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2022

Keywords

Crossrefs

Cf. A001818, A067856, A353819, A353914, A353972, A354116, A354172, A354275, A354276.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = {1, 0, -1, 0}[[Mod[n, 4, 1]]] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arcsinh(x^k)) / k.

A354275 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctan(x).

Original entry on oeis.org

1, 0, -2, 8, -16, -64, -832, 13824, 8192, -36096, -4228608, -58438656, -398991360, -3452915712, 44581613568, 7144463302656, -17762113880064, 126440605483008, -7331825098948608, -88237584523984896, 3154526750647517184, -27279757707305287680, -1278044473427380666368
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2022

Keywords

Crossrefs

Cf. A010050, A067856, A353820, A353915, A353972, A354117, A354175, A354274, A354276.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = {1, 0, -1, 0}[[Mod[n, 4, 1]]]/n - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arctan(x^k)) / k.

A354276 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arctanh(x).

Original entry on oeis.org

1, 0, 2, -8, 64, -304, 3968, -43392, 378880, -4002816, 68247552, -995736576, 15949529088, -238273241088, 4760383438848, -113132156780544, 2119956936523776, -42743492966350848, 1123874181449515008, -28901050300546154496, 722523072906903158784, -19401957422023594475520, 589068777481530305937408
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2022

Keywords

Crossrefs

Cf. A010050, A067856, A353821, A353928, A353972, A354118, A354176, A354274, A354275.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = Mod[n, 2]/n - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arctanh(x^k)) / k.

A361987 a(1) = 1; a(n) = n^2 * Sum_{d|n, d < n} (-1)^(n/d) a(d) / d^2.

Original entry on oeis.org

1, 4, -9, 32, -25, -36, -49, 256, 0, -100, -121, -288, -169, -196, 225, 2048, -289, 0, -361, -800, 441, -484, -529, -2304, 0, -676, 0, -1568, -841, 900, -961, 16384, 1089, -1156, 1225, 0, -1369, -1444, 1521, -6400, -1681, 1764, -1849, -3872, 0, -2116, -2209, -18432, 0, 0, 2601, -5408, -2809, 0
Offset: 1

Views

Author

Seiichi Manyama, Apr 02 2023

Keywords

Links

Crossrefs

Partial sums give A361983.
Cf. A067856, A332793.
Cf. A334657, A361986.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := 2^(3*e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 09 2023 *)

Formula

a(n) is multiplicative with a(2^e) = 2^(3*e-1). a(p) = -p^2, a(p^e) = 0 if e>1, p>2.
G.f. A(x) satisfies -x = Sum_{k>=1} (-1)^k * k^2 * A(x^k).

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, -2, 2, 1, 2, -3, 1, -3, -1, -7, 8, 4, 9, -7, 7, -7, 0, -21, 15, 2, 18, -23, 8, -25, -1, -43, 46, 17, 58, -34, 40, -41, 9, -98, 79, 10, 100, -98, 40, -123, -2, -191, 176, 43, 237, -136, 144, -192, 30, -362, 277, 12, 373, -314, 131, -457, -9, -606
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2018

Keywords

Links

Crossrefs

Cf. A067856, A073707, A129373, A321326.

Formula

G.f.: Product_{k>0} (1 + x^k)^A067856(k).
Product_{k>0} A(x^k) = Product_{k>=0} (1 + x^(2^k))^(2^k). (Cf. A073707.)

A327274 Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 2^(1 - s))).

Original entry on oeis.org

1, 0, -2, 1, -2, 0, -2, 2, 1, 0, -2, -2, -2, 0, 4, 4, -2, 0, -2, -2, 4, 0, -2, -4, 1, 0, 0, -2, -2, 0, -2, 8, 4, 0, 4, 1, -2, 0, 4, -4, -2, 0, -2, -2, -2, 0, -2, -8, 1, 0, 4, -2, -2, 0, 4, -4, 4, 0, -2, 4, -2, 0, -2, 16, 4, 0, -2, -2, 4, 0, -2, 2, -2, 0, -2, -2, 4, 0, -2, -8
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

Comments

Dirichlet inverse of A048272.
Moebius transform of A067856.

Links

Crossrefs

Cf. A007427, A008683, A048272, A062503 (positions of 1's), A067856, A327268.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Sum[Sum[(-1)^j, {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 80}]
f[p_, e_] := Switch[e, 1, -2, 2, 1, , 0]; f[2, e] := 2^(e-2); f[2, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)
  • PARI
    A067856(n) = { my(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1)); }; \\ From A067856
A327274(n) = sumdiv(n,d,moebius(n/d)*A067856(d));

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA048272(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A067856(d).
a(n) = 0 if n == 2 (mod 4). - Bernard Schott, Dec 07 2021
Multiplicative with a(2) = 0, a(2^e) = 2^(e-2) for e >= 2, and for an odd prime p, a(p) = -2, a(p^2) = 1, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023

A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).

Original entry on oeis.org

1, -1, 2, -3, 3, -4, 8, -9, 6, -9, 14, -15, 16, -17, 27, -33, 21, -22, 36, -37, 34, -45, 61, -62, 51, -55, 73, -82, 76, -77, 124, -125, 80, -97, 120, -132, 132, -133, 171, -190, 153, -154, 221, -222, 194, -233, 296, -297, 239, -248, 313, -337, 301, -302
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 10 2022

Keywords

Crossrefs

Cf. A003238, A067856, A281487, A307776, A307778, A308077, A325144, A343370.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1,
      add((-1)^((n-1)/d)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..54);  # Alois P. Heinz, Feb 10 2022
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 54}]
nmax = 54; A[] = 0; Do[A[x] = x (1 + Sum[(-1)^k A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * ( 1 - A(x) + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... ).
G.f.: x * ( 1 - Sum_{n>=1} a(n) * x^n / (1 + x^n) ).
Previous Showing 21-29 of 29 results.

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