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 21-30 of 30 results.

A336998 a(n) = n! * Sum_{d|n} 3^(d - 1) / d!.

Original entry on oeis.org

1, 5, 15, 87, 201, 3123, 5769, 148347, 913761, 11541123, 39975849, 2616723387, 6227552241, 230557039443, 4151870901369, 76980002233707, 355687471142721, 27886053280896963, 121645100796252489, 10474674957482235867, 135117295282596928401, 2811664555920692775603
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

Links

Crossrefs

Cf. A034730, A053486, A057625, A336997.

Programs

  • Magma
    A336998:= func< n | Factorial(n)*(&+[3^(d-1)/Factorial(d): d in Divisors(n)]) >;
[A336998(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    Table[n! Sum[3^(d - 1)/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[(Exp[3 x^k] - 1)/3, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sumdiv(n, d, 3^(d-1)/d!); \\ Michel Marcus, Aug 12 2020
  • SageMath
    def A336998(n): return factorial(n)*sum(3^(k-1)/factorial(k) for k in (1..n) if (k).divides(n))
[A336998(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

E.g.f.: Sum_{k>=1} (exp(3*x^k) - 1) / 3.
a(p) = p! + 3^(p - 1), where p is prime.

A354862 a(n) = n! * Sum_{d|n} (n/d)! / d!.

Original entry on oeis.org

1, 5, 37, 601, 14401, 520801, 25401601, 1626189601, 131682257281, 13168407228481, 1593350922240001, 229442707280223361, 38775788043632640001, 7600054676241325858561, 1710012252750418295078401, 437763137119219420513804801, 126513546505547170185216000001
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2022

Keywords

Crossrefs

Cf. A020549, A057625, A121860, A354843, A354863.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, (n/#)! / #! &]; Array[a, 17] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = n!*sumdiv(n, d, (n/d)!/d!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k!*(exp(x^k)-1))))
  • Python
    from math import factorial
from sympy import divisors
def A354862(n):
    f = factorial(n)
    return sum(f*(a := factorial(n//d))//(b:= factorial(d)) + (f*b//a if d**2 < n else 0) for d in divisors(n,generator=True) if d**2 <= n) # Chai Wah Wu, Jun 09 2022

Formula

E.g.f.: Sum_{k>0} k! * (exp(x^k) - 1).
If p is prime, a(p) = 1 + (p!)^2 = A020549(p).

A329966 a(n) = n! * Sum_{d|n} binomial(n-1,d-1) / d!.

Original entry on oeis.org

1, 3, 7, 61, 121, 3721, 5041, 240241, 2056321, 23768641, 39916801, 11104853761, 6227020801, 683519316481, 32048919302401, 577844178912001, 355687428096001, 261396772808371201, 121645100408832001, 202418558674082150401, 2061884451929702400001, 12935940353987812761601
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

Links

Crossrefs

Cf. A000262, A056045, A057625, A271654.

Programs

  • Magma
    [Factorial(n)*( &+[ Binomial(n-1,d-1)/Factorial(d):d in Divisors(n)]): n in [1..22]]; // Marius A. Burtea, Jan 02 2020
  • Maple
    N:= 30:
V:= Vector(N):
for d from 1 to N do
   for k from 1 to floor(N/d) do
     n:= k*d; V[n]:= V[n] + n!/d!*binomial(n-1,d-1);
od od:
convert(V,list); # Robert Israel, Jan 01 2020
  • Mathematica
    a[n_] := n! Sum[Binomial[n - 1, d - 1]/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
  • PARI
    a(n) = n! * sumdiv(n, d, binomial(n-1,d-1) / d!); \\ Michel Marcus, Nov 26 2019

A336241 a(n) = (n!)^2 * Sum_{d|n} 1 / (d!)^2.

Original entry on oeis.org

1, 5, 37, 721, 14401, 662401, 25401601, 2034950401, 135339724801, 16461151257601, 1593350922240001, 293575350020198401, 38775788043632640001, 9500068369885892198401, 1757631343928533032960001, 547963926586675321282560001, 126513546505547170185216000001
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 13 2020

Keywords

Crossrefs

Cf. A006040, A057625, A336242.

Programs

  • Mathematica
    Table[(n!)^2 Sum[1/(d!)^2, {d, Divisors[n]}], {n, 1, 17}]
nmax = 17; CoefficientList[Series[Sum[(BesselI[0, 2 x^(k/2)] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2 // Rest
  • PARI
    a(n) = n!^2*sumdiv(n, d, 1/d!^2); \\ Michel Marcus, Jul 13 2020

Formula

a(n) = (n!)^2 * [x^n] Sum_{k>=1} (BesselI(0,2*x^(k/2)) - 1).
a(n) = (n!)^2 * [x^n] Sum_{k>=1} x^k / ((k!)^2 * (1 - x^k)).

A336997 a(n) = n! * Sum_{d|n} 2^(d - 1) / d!.

Original entry on oeis.org

1, 4, 10, 56, 136, 1952, 5104, 94208, 605056, 7741952, 39917824, 1458295808, 6227024896, 175463616512, 2353813878784, 48886264659968, 355687428161536, 17362063156969472, 121645100409094144, 6001501553433509888, 85800344155030552576, 2248030289949388439552
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

Crossrefs

Cf. A010842, A034729, A057625, A336998.

Programs

  • Mathematica
    Table[n! Sum[2^(d - 1)/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[(Exp[2 x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sumdiv(n, d, 2^(d-1)/d!); \\ Michel Marcus, Aug 12 2020

Formula

E.g.f.: Sum_{k>=1} (exp(2*x^k) - 1) / 2.
a(p) = p! + 2^(p - 1), where p is prime.

A354022 a(n) = n! * Sum_{d|n} mu(n/d) / d!.

Original entry on oeis.org

1, -1, -5, -11, -119, 241, -5039, -1679, -60479, 1784161, -39916799, 218877121, -6227020799, 43571848321, 1078831353601, -518918399, -355687428095999, 1058152455360001, -121645100408831999, 1115079416638387201, 42565648051390464001, 562000335730215782401
Offset: 1

Views

Author

Ilya Gutkovskiy, May 14 2022

Keywords

Links

Crossrefs

Cf. A008683, A057625, A062794, A068107, A099740, A132958, A327243, A332466.

Programs

  • Mathematica
    Table[n! Sum[MoebiusMu[n/d]/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] (Exp[x^k] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n)=n! * sumdiv(n, d, moebius(n/d) / d!) \\ Winston de Greef, Sep 19 2023

Formula

E.g.f.: Sum_{k>=1} mu(k) * (exp(x^k) - 1).
Sum_{n>=1} a(n) * x^n / (n! * (1 - x^n)) = exp(x) - 1.

A354900 a(n) = n! * Sum_{d|n} d^d / (n/d)!.

Original entry on oeis.org

1, 9, 163, 6193, 375001, 33602521, 4150656721, 676462516801, 140587148681281, 36288005670120961, 11388728893445164801, 4270826391670469473921, 1886009588552176549862401, 968725766890781857146309121, 572622616354852243874626732801
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2022

Keywords

Crossrefs

Cf. A057625, A354843, A354888, A354892, A354899.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^#/(n/#)! &]; Array[a, 15] (* Amiram Eldar, Jun 11 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^d/(n/d)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k^k*(exp(x^k)-1))))

Formula

E.g.f.: Sum_{k>0} k^k * (exp(x^k) - 1).
If p is prime, a(p) = 1 + p^p * p!.

A330020 Expansion of e.g.f. Sum_{k>=1} x^k / (k! * (1 - x^k)^k).

Original entry on oeis.org

1, 3, 7, 49, 121, 2161, 5041, 127681, 725761, 12852001, 39916801, 2917918081, 6227020801, 392423391361, 4740319584001, 122053759027201, 355687428096001, 57808258040332801, 121645100408832001, 18854997267794688001, 289799177540640768001, 7306005040298918553601
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

Crossrefs

Cf. A057625, A157019, A327578, A327579.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Sum[x^k/(k! (1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := n! Sum[(d + n/d - 2)!/(d! (d - 1)! (n/d - 1)!), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]

Formula

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

A336999 a(n) = n! * Sum_{d|n} n^d / d!.

Original entry on oeis.org

1, 8, 45, 544, 3725, 89856, 858823, 25271296, 434776329, 13241728000, 285750755411, 11494661861376, 302956057862653, 12945137688641536, 446924199188379375, 20735627677666902016, 827246308572614396177, 43155924331583693389824
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 10 2020

Keywords

Crossrefs

Cf. A057625, A063170, A066108.

Programs

  • Mathematica
    Table[n! Sum[n^d/d!, {d, Divisors[n]}], {n, 1, 18}]
Table[n! SeriesCoefficient[Sum[(Exp[n x^k] - 1), {k, 1, n}], {x, 0, n}], {n, 1, 18}]
  • PARI
    a(n) = n! * sumdiv(n, d, n^d/d!); \\ Michel Marcus, Aug 12 2020

Formula

a(n) = n! * [x^n] Sum_{k>=1} (exp(n*x^k) - 1).

A357296 Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).

Original entry on oeis.org

1, 3, 7, 31, 121, 851, 5041, 43261, 369601, 3748249, 39916801, 490801081, 6227020801, 87861842641, 1310800947457, 21018206008801, 355687428096001, 6419518510204801, 121645100408832001, 2435836129700029057, 51102829650622464001, 1124549558817839481601
Offset: 1

Views

Author

Seiichi Manyama, Feb 23 2023

Keywords

Crossrefs

Cf. A038507, A057625, A327578, A354891.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, 1/(#^(n/#-1) * #!) &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k/k)))))
  • PARI
    a(n) = n!*sumdiv(n, d, 1/(d^(n/d-1)*d!));

Formula

a(n) = n! * Sum_{d|n} 1 / (d^(n/d-1) * d!).
If p is prime, a(p) = 1 + p! = A038507(p).
Previous Showing 21-30 of 30 results.

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