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 11-17 of 17 results.

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Original entry on oeis.org

1, 3, 1, 16, 1, 125, 15, 1, 1296, 1, 16807, 735, 140, 1, 262144, 1, 4782969, 76545, 1890, 1, 100000000, 112000, 1, 2357947691, 13835745, 33264, 1, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375
Offset: 1

Views

Author

Gus Wiseman, Jul 03 2019

Keywords

Comments

A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.

Examples

			Triangle begins:
           1
           3          1
          16          1
         125         15          1
        1296          1
       16807        735        140          1
      262144          1
     4782969      76545       1890          1
   100000000     112000          1
  2357947691   13835745      33264          1
The T(4,2) = 15 hypertrees:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}

Links

Crossrefs

Row sums are A320444.
Column d = 1 is A000272.
Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

Programs

  • Maple
    T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 21 2019
  • Mathematica
    Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]

Formula

T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).

Extensions

Edited by Peter Munn, Mar 05 2025

A356662 a(n) = n! * Sum_{d|n} 1/(d!)^(n/d - 1).

Original entry on oeis.org

1, 4, 12, 60, 240, 1740, 10080, 87360, 735840, 7514640, 79833600, 976686480, 12454041600, 175736040480, 2616448554720, 42011071502400, 711374856192000, 12830610027755520, 243290200817664000, 4870565189425615680, 102182981410948838400, 2249099140674523737600
Offset: 1

Views

Author

Seiichi Manyama, Aug 21 2022

Keywords

Crossrefs

Cf. A061095, A098558, A356543, A356661.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, 1/(#!)^(n/# - 1) &]; Array[a, 22] (* Amiram Eldar, Aug 21 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, 1/d!^(n/d-1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(1-x^k/k!))))

Formula

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

A327587 a(n) = n! * Sum_{d|n} (-1)^(n - d) / (n/d)!^d.

Original entry on oeis.org

1, 1, 7, 29, 121, 649, 5041, 42909, 364561, 3515651, 39916801, 486821873, 6227020801, 86497214231, 1307843292757, 21004582611869, 355687428096001, 6390006277567483, 121645100408832001, 2435277595236694779, 51091124681475552961, 1123451899297248225431
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 17 2019

Keywords

Crossrefs

Cf. A061095, A327243.

Programs

  • Mathematica
    a[n_] := n! Sum[(-1)^(n - d)/(n/d)!^d, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[-(-x)^k/(k! + (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n!*sumdiv(n, d, (-1)^(n-d)/(n/d)!^d); \\ Michel Marcus, Sep 19 2019

Formula

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

A332467 a(n) = n! * Sum_{d|n} mu(d) / (d!)^(n/d).

Original entry on oeis.org

1, 1, 5, 18, 119, 611, 5039, 37800, 361200, 3515149, 39916799, 471148524, 6227020799, 86497207369, 1307505443245, 20841060240000, 355687428095999, 6389731861649136, 121645100408831999, 2430526115576719732, 51090759661943327041, 1123451899297246814569
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Crossrefs

Cf. A008683, A061095, A327587, A332466.

Programs

  • Magma
    [Factorial(n)*&+[MoebiusMu(d) /(Factorial(d))^(n div d):d in Divisors(n)]:n in [1..22]]; // Marius A. Burtea, Feb 13 2020
  • Mathematica
    Table[n! DivisorSum[n, MoebiusMu[#]/(#!)^(n/#) &], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(k! - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n)={sumdiv(n, d, moebius(d)*n!/(d!)^(n/d))} \\ Andrew Howroyd, Feb 13 2020

Formula

E.g.f.: Sum_{k>=1} Sum_{j>=1} mu(j) * x^(k*j) / (j!)^k.
E.g.f.: Sum_{k>=1} mu(k) * x^k / (k! - x^k).
a(n) ~ n!. - Vaclav Kotesovec, Feb 16 2020

A355669 a(n) = n! * Sum_{d|n} (d!)^(d - n/d).

Original entry on oeis.org

1, 6, 222, 331824, 24883200120, 139314069504005400, 82606411253903523840005040, 6984964247141514123629140377623274720, 109110688415571316480344899355894085582848000725760, 395940866122425193243875570782668457763038822400000006270570482400
Offset: 1

Views

Author

Seiichi Manyama, Aug 21 2022

Keywords

Crossrefs

Cf. A061095, A345465, A351165, A356662.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, (#!)^(# - n/#) &]; Array[a, 10] (* Amiram Eldar, Aug 21 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!*x)^k/(k!-x^k))))

Formula

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

A358593 a(n) = n! * Sum_{d|n} d^(n-d) / d!^(n/d).

Original entry on oeis.org

1, 3, 7, 49, 121, 2701, 5041, 219521, 1587601, 33446701, 39916801, 17731796545, 6227020801, 2879710009177, 98069239768501, 2418218838097921, 355687428096001, 2832293713653708877, 121645100408832001, 2295597943489176040001, 71029619657111138063041
Offset: 1

Views

Author

Seiichi Manyama, Feb 23 2023

Keywords

Crossrefs

Cf. A038507, A061095, A354891, A356668.

Programs

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

Formula

E.g.f.: Sum_{k>0} x^k / (k! - (k * x)^k).
If p is prime, a(p) = 1 + p! = A038507(p).

A358594 Expansion of e.g.f. Sum_{k>0} x^k / ((k-1)! - x^k).

Original entry on oeis.org

1, 4, 9, 52, 125, 1626, 5047, 81768, 408249, 7263910, 39916811, 990191676, 6227020813, 174356750582, 1348633786515, 41861724727696, 355687428096017, 12904788209623614, 121645100408832019, 4866124222538035620, 51490090294307945301, 2248001455555300717294
Offset: 1

Views

Author

Seiichi Manyama, Feb 23 2023

Keywords

Crossrefs

Cf. A061095, A356668.

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, 1/(#-1)!^(n/#) &]; 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))))
  • PARI
    a(n) = n!*sumdiv(n, d, 1/(d-1)!^(n/d));

Formula

a(n) = n! * Sum_{d|n} 1/(d-1)!^(n/d).
If p is prime, a(p) = p + p!.
Previous Showing 11-17 of 17 results.

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