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 1-10 of 10 results.

A354892 a(n) = n! * Sum_{d|n} d^n / (n/d)!.

Original entry on oeis.org

1, 9, 163, 6337, 375001, 33862441, 4150656721, 677778984961, 140588337476161, 36305718780965761, 11388728893445164801, 4271349071581227377281, 1886009588552176549862401, 968755330019156299208709121, 572622623006183707899105964801
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2022

Keywords

Links

Crossrefs

Cf. A354843, A354890.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^n/(n/#)! &]; Array[a, 15] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^n/(n/d)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp((k*x)^k)-1)))

Formula

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

A354845 a(n) = n! * Sum_{d|n} (n/d)^(d-1) / d!.

Original entry on oeis.org

1, 3, 7, 49, 121, 2281, 5041, 134401, 907201, 13184641, 39916801, 3753509761, 6227020801, 393409336321, 7638997766401, 160474477363201, 355687428096001, 75792615407308801, 121645100408832001, 32459310892353945601, 475723576423839744001, 7306033564948620902401
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2022

Keywords

Links

Crossrefs

Cf. A087906, A327578, A354843.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, (n/#)^(#-1)/#! &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, (n/d)^(d-1)/d!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(k*x^k)-1)/k)))

Formula

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

A354888 a(n) = n! * Sum_{d|n} d^d / d!.

Original entry on oeis.org

1, 6, 33, 328, 3245, 52056, 828583, 17328256, 389416329, 10105386400, 285351587411, 8955841614336, 302881333613053, 11126513414294656, 437935136609883375, 18455736024587862016, 827240617573764860177, 39353706314004951028224
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2022

Keywords

Links

Crossrefs

Cf. A327579, A354843, A354889.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^#/#! &]; Array[a, 18] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^d/d!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (k*x)^k/(k!*(1-x^k)))))

Formula

E.g.f.: Sum_{k>0} (k * x)^k/(k! * (1 - x^k)).
If p is prime, a(p) = p^p + p! = A053042(p).

A327579 a(n) = n! * Sum_{d|n} d^(n/d) / d!.

Original entry on oeis.org

1, 4, 9, 76, 125, 4686, 5047, 389768, 1995849, 62445610, 39916811, 23574862092, 6227020813, 5667436494734, 55630647072015, 2922249531801616, 355687428096017, 2425220588831040018, 121645100408832019, 1364553980880330240020, 18677216386213152768021, 1152100749379237026969622
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 17 2019

Keywords

Links

Crossrefs

Cf. A055225, A057625, A327578, A354843.

Programs

  • Mathematica
    a[n_] := n! Sum[d^(n/d)/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/((k - 1)! (1 - k x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sumdiv(n, d, d^(n/d) / d!); \\ Michel Marcus, Sep 17 2019

Formula

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

A354863 a(n) = n! * Sum_{d|n} (n/d) / d!.

Original entry on oeis.org

1, 5, 19, 121, 601, 5641, 35281, 406561, 3447361, 45420481, 439084801, 7565564161, 80951270401, 1525654690561, 20737536019201, 421943967244801, 6046686277632001, 150482493928166401, 2311256907767808001, 61410502863943833601, 1132546296081328128001
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2022

Keywords

Crossrefs

Cf. A057625, A354843, A354862.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, (n/#) / #! &]; Array[a, 21] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = n!*sumdiv(n, d, n/d/d!);
  • PARI
    my(N=30, 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 A354863(n):
    f = factorial(n)
    return sum(f*n//d//factorial(d) for d in divisors(n,generator=True)) # 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 * p!.

A354844 a(n) = n! * Sum_{d|n} (n/d)^d / (d! * (n/d)!).

Original entry on oeis.org

1, 3, 4, 29, 6, 1027, 8, 26889, 272170, 861851, 12, 515592013, 14, 1530809295, 668366899216, 9382044672017, 18, 1405750464518419, 20, 1393382139935385621, 4274473667143680022, 30537988748467223, 24, 211745638285336995840025
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2022

Keywords

Crossrefs

Cf. A121860, A354843, A354845.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, (n/#)^#/(#! * (n/#)!) &]; Array[a, 25] (* Amiram Eldar, Jun 08 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, (n/d)^d/(d!*(n/d)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(k*x^k)-1)/k!)))

Formula

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

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).

A363697 a(n) = -n! * Sum_{d|n} (-n/d)^d / d!.

Original entry on oeis.org

1, 3, 19, 47, 601, 2039, 35281, -26881, 4898881, -8104321, 439084801, 576132479, 80951270401, -913158005761, 49506372115201, -558073906790401, 6046686277632001, 79958674981785599, 2311256907767808001, -115583806104986419201
Offset: 1

Views

Author

Seiichi Manyama, Jun 16 2023

Keywords

Links

Crossrefs

Cf. A132958, A363698.
Cf. A354843.

Programs

  • Mathematica
    a[n_] := -n! * DivisorSum[n, (-n/#)^#/#! &]; Array[a, 20] (* Amiram Eldar, Jul 03 2023 *)
  • PARI
    a(n) = -n!*sumdiv(n, d, (-n/d)^d/d!);

Formula

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

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!.

A356494 Expansion of e.g.f. Product_{k>0} B(k * x^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 6, 35, 327, 2892, 37943, 459895, 7330172, 116054835, 2168292295, 41072348550, 898738816957, 19782331776937, 487091519709590, 12305361661242275, 337777113607935587, 9528258228302443724, 289373132780801591323, 9016757353084706862647
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2022

Keywords

Crossrefs

Cf. A209903, A346055, A356495.
Cf. A000110, A354843, A356460.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(k*x^k)-1))))
  • PARI
    a354843(n) = n!*sumdiv(n, d, (n/d)^d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354843(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354843(k) * binomial(n-1,k-1) * a(n-k).
Showing 1-10 of 10 results.

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