A057625 -id:A057625 - cp's OEIS frontend

A330199 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k)).

1, -1, -2, 1, 1, 98, -39, 3225, 1226, 6459, 12473, 821830, -214739887, -201448561, -8997850614, -514986723363, -1310942141971, -26465356716946, -931753364233567, -1858534483400559, 167210272584038942, -7112146717031426801, 312288595642509829797

Offset: 0

Ilya Gutkovskiy, Dec 05 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000587, A057625, A209903.

nmax = 22; CoefficientList[Series[Product[Exp[1 - Exp[x^k]], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] k! DivisorSum[k, 1/#! &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of complementary Bell numbers (A000587).
E.g.f.: exp(-Sum_{j>=1} Sum_{i>=1} x^(i*j) / i!).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A057625(k) * a(n-k).

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

Ilya Gutkovskiy, Sep 17 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..425

Cf. A055225, A057625, A327578, A354843.

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

a(n) = n! * sumdiv(n, d, d^(n/d) / d!); \\ Michel Marcus, Sep 17 2019

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

A330254 Expansion of e.g.f. Sum_{k>=1} sinh(x^k).

Original entry on oeis.org

1, 2, 7, 24, 121, 840, 5041, 40320, 423361, 3659040, 39916801, 558835200, 6227020801, 87195588480, 1536517382401, 20922789888000, 355687428096001, 7469453633241600, 121645100408832001, 2453176191578112000, 59616236292028416001

Offset: 1

Ilya Gutkovskiy, Dec 07 2019

nonn

Cf. A057625, A132958, A176473, A330255.

nmax = 21; CoefficientList[Series[Sum[Sinh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! DivisorSum[n, 1/#! &, OddQ[#] &], {n, 1, 21}]

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

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

A330255 Expansion of e.g.f. Sum_{k>=1} (cosh(x^k) - 1) (even powers only).

Original entry on oeis.org

1, 13, 361, 21841, 1814401, 260124481, 43589145601, 11333696774401, 3210079038566401, 1317822591538252801, 562000363888803840001, 336953340897297630105601, 201645730563302817792000001, 165147853334842304408401920001, 132994909752412012763531673600001

Offset: 1

Ilya Gutkovskiy, Dec 07 2019

nonn

Cf. A057625, A132958, A176474, A330254.

nmax = 15; Table[(CoefficientList[Series[Sum[Cosh[x^k] - 1, {k, 1, nmax}], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}] // Rest
Table[(2 n)! DivisorSum[n, 1/(2 #)! &], {n, 1, 15}]

E.g.f.: Sum_{k>=1} x^(2*k) / ((2*k)! * (1 - x^(2*k))) (even powers only).

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

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

Seiichi Manyama, Jun 09 2022

nonn

Cf. A057625, A354843, A354862.

a[n_] := n! * DivisorSum[n, (n/#) / #! &]; Array[a, 21] (* Amiram Eldar, Aug 30 2023 *)

PARI
```
a(n) = n!*sumdiv(n, d, n/d/d!);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*(exp(x^k)-1))))

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

E.g.f.: Sum_{k>0} k * (exp(x^k) - 1).

If p is prime, a(p) = 1 + p * p!.

A087905 a(n) = n! * Sum_{d|n} (d/n)^d.

Original entry on oeis.org

1, 3, 8, 36, 144, 1010, 5760, 50400, 416640, 4250232, 43545600, 553106400, 6706022400, 95865541200, 1410695430144, 22720842144000, 376610217984000, 6888030445296000, 128047474114560000, 2587520533615041024

Offset: 1

Vladeta Jovovic, Oct 14 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..445

Cf. A061095, A038041, A057625, A038048, A005225.

a[n_]:= n!*DivisorSum[n, (#/n)^# &]; Array[a, 50] (* G. C. Greubel, May 16 2018 *)

{a(n)= n!*sumdiv(n, d, (d/n)^d)};
for(n=1, 30, print1(a(n), ", ")) \\ G. C. Greubel, May 16 2018

E.g.f.: Sum_{k>0} x^k/(k-x^k).

A327243 a(n) = n! * Sum_{d|n} (-1)^(n - d) / (n/d)!.

Original entry on oeis.org

1, 1, 7, 35, 121, 479, 5041, 62159, 423361, 1844639, 39916801, 779042879, 6227020801, 43606442879, 1536517382401, 32256486662399, 355687428096001, 4259374594675199, 121645100408832001, 3568256949101644799, 59616236292028416001, 562000392047391897599

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

nonn

Cf. A057625, A132958.

[Factorial(n)*(&+[(-1)^(n-d)/Factorial(n div d):d in Divisors(n)]):n in [1..22]]; // Marius A. Burtea, Sep 14 2019

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

a(n) = {n!*sumdiv(n, d, (-1)^(n - d) / (n/d)!)} \\ Andrew Howroyd, Sep 14 2019

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

E.g.f.: Sum_{k>=1} (-1)^k * (exp((-x)^k) - 1). [corrected by Ilya Gutkovskiy, May 14 2022]

A066387 Triangle T(n,m) (1<=m<=n) giving number of maps f:N -> N such that f^m(X)=X+n for all natural numbers X.

Original entry on oeis.org

1, 1, 2, 1, 0, 6, 1, 12, 0, 24, 1, 0, 0, 0, 120, 1, 120, 360, 0, 0, 720, 1, 0, 0, 0, 0, 0, 5040, 1, 1680, 0, 20160, 0, 0, 0, 40320, 1, 0, 60480, 0, 0, 0, 0, 0, 362880, 1, 30240, 0, 0, 1814400, 0, 0, 0, 0, 3628800, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800

Offset: 1

Floor van Lamoen, Dec 23 2001

			Triangle T(n,m) begins:
  1;
  1,    2;
  1,    0,   6;
  1,   12,   0,    24;
  1,    0,   0,     0, 120;
  1,  120, 360,     0,   0, 720;
  1,    0,   0,     0,   0,   0, 5040;
  1, 1680,   0, 20160,   0,   0,    0, 40320;
  ...

Vincenzo Librandi, Rows n = 1..100, flattened
A. Heinis, R. Jeurissen and L. Kamstra, Problem 18 and solution, Nieuw Arch. Wisk. 5/2 (2001) 380.

Row sums give A057625.
Main diagonal gives A000142.
m-section of column m=2-4 (for n>0) gives: A001813, A064350, A166338.

t[n_, m_] /; Divisible[n, m] := n!/(n/m)!; t[, ] = 0; Flatten[Table[t[n, m], {n, 1, 11}, {m, 1, n}]] (* Jean-François Alcover, Nov 29 2011 *)

T(n,m) = n!/(n/m)! if m|n, T(n,m) = 0 otherwise.

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

Gus Wiseman, Jul 03 2019

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.

			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}}

Alois P. Heinz, Rows n = 1..185, flattened

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

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

Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]

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

Edited by Peter Munn, Mar 05 2025

A332466 a(n) = n! * Sum_{d|n} mu(d) / d!.

Original entry on oeis.org

1, 1, 5, 12, 119, 241, 5039, 20160, 302400, 1784161, 39916799, 160332480, 6227020799, 43571848321, 1078831353601, 10461394944000, 355687428095999, 2143016754278400, 121645100408831999, 1196177491129420800, 42565648051390464001, 562000335730215782401

Offset: 1

Ilya Gutkovskiy, Feb 13 2020

nonn

Cf. A008683, A057625, A068107, A099740, A132958, A332467.

with(numtheory):
a:= n-> n! * add(mobius(d)/d!, d=divisors(n)):
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 13 2020

Table[n! DivisorSum[n, MoebiusMu[#]/#! &], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] x^k/(k! (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n)={sumdiv(n, d, moebius(d)*n!/d!)} \\ Andrew Howroyd, Feb 13 2020

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

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

cp's OEIS Frontend

A330199 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k)).

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

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A330254 Expansion of e.g.f. Sum_{k>=1} sinh(x^k).

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A330255 Expansion of e.g.f. Sum_{k>=1} (cosh(x^k) - 1) (even powers only).

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

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A087905 a(n) = n! * Sum_{d|n} (d/n)^d.

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A327243 a(n) = n! * Sum_{d|n} (-1)^(n - d) / (n/d)!.

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A066387 Triangle T(n,m) (1<=m<=n) giving number of maps f:N -> N such that f^m(X)=X+n for all natural numbers X.

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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).

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