A258903 -id:A258903 - cp's OEIS frontend

A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

1, 2, 2, 8, 2, 122, 2, 1682, 10082, 30242, 2, 7318082, 2, 17297282, 3632428802, 36843206402, 2, 2981705126402, 2, 1690185726028802, 3379030566912002, 28158588057602, 2, 76941821303636889602, 1077167364120207360002

Offset: 1

Vladeta Jovovic, Sep 09 2006

a(n) = 2 iff n is prime.
a(468) has 1007 decimal digits. - Michael De Vlieger, Sep 12 2018
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to row and column permutations. For example, inequivalent representatives of the a(4) = 8 matrices are:
  [1 2 3 4]
.
  [1 2] [1 2] [1 3] [1 3] [1 4] [1 4]
  [3 4] [4 3] [2 4] [4 2] [2 3] [3 2]
.
  [1]
  [2]
  [3]
  [4]
(End)
Conjecture: the sequence a(n) taken modulo a positive integer k >= 3 eventually becomes constant equal to 2. For example, the sequence taken modulo 11 is [1, 2, 2, 8, 2, 1, 2, 10, 6, 3, 2, 2, 2, 2, 2, 2, ...]. - Peter Bala, Aug 08 2025

Michael De Vlieger, Table of n, a(n) for n = 1..467
Jimmy Devillet, Gergely Kiss, Characterizations of biselective operations, arXiv:1806.02073 [math.RA], 2018.

Cf. A038041, A057625, A061095, A100195, A236696, A258899, A258903, A320444, A386877.

with(numtheory): seq(n!*add(1/(d!*(n/d)!), d in divisors(n)), n = 1..25); # Peter Bala, Aug 04 2025

f[n_] := Block[{d = Divisors@n}, Plus @@ (n!/(d! (n/d)!))]; Array[f, 25] (* Robert G. Wilson v, Sep 11 2006 *)
Table[DivisorSum[n, n!/(#!*(n/#)!) &], {n, 25}] (* Michael De Vlieger, Sep 12 2018 *)

a(n) = sumdiv(n, d, n!/(d!*(n/d)!)); \\ Michel Marcus, Sep 13 2018

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

More terms from Robert G. Wilson v, Sep 11 2006

A258899 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.

Original entry on oeis.org

1, 2, 6, 10, 42, 34, 786, 130, 17058, 81154, 545346, 2050, 102457218, 8194, 1141636866, 72648608770, 648648065538, 131074, 111258180895746, 524290, 40892974286411778, 229774078552113154, 28890711351291906, 8388610, 3552178288049960329218, 34469355651846669074434

Offset: 0

Paul D. Hanna, Jun 20 2015

nonn

Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k). For example, the sequence taken modulo 11 is [1, 2, 6, 10, 9, 1, 5, 9, 8, 7, 10, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, 4, 6, 10, 7, 1, 0, 9, 5, 8, 3, ...] with an apparent period of 10 (= phi(11)) starting at n = 11. - Peter Bala, Aug 03 2025

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 10*x^3/3! + 42*x^4/4! + 34*x^5/5! + 786*x^6/6! +...
where
A(x) = 2 - exp(2) + 2*exp(x) + 2^2*exp(x^2)/2! + 2^3*exp(x^3)/3! + 2^4*exp(x^4)/4! + 2^5*exp(x^5)/5! +...
A(x) = 2 - exp(1) + exp(2*x) + exp(2*x^2)/2! + exp(2*x^3)/3! + exp(2*x^4)/4! + exp(2*x^5)/5! +...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A121860, A258903.

with(numtheory): seq(`if`(n=0, 1, n!*add(2^d/(d!*(n/d)!), d in divisors(n))), n = 0..25); # Peter Bala, Aug 04 2025

{a(n) = local(A=1); A = 2-exp(2) + sum(m=1,n,2^m/m!*exp(x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))

{a(n) = local(A=1); A = 2-exp(1) + sum(m=1,n,1/m!*exp(2*x^m +x*O(x^n))); if(n==0,1, n!*polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))

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

For n >= 1, a(n) = Sum_{d divides n} 2^d * n!/(d!*(n/d)!). - Peter Bala, Aug 04 2025

cp's OEIS Frontend

A121860 a(n) = Sum_{d|n} n!/(d!*(n/d)!).

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A258899 E.g.f.: 2 - exp(2) + Sum_{n>=1} 2^n * exp(x^n) / n!.

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