A038048 -id:A038048 - cp's OEIS frontend

A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

1, 3, 7, 37, 121, 1201, 5041, 62161, 423361, 5473441, 39916801, 818959681, 6227020801, 130784734081, 1536517382401, 32256486662401, 355687428096001, 10679532671808001, 121645100408832001, 3770998783116364801, 59616236292028416001, 1686001119824999577601

Offset: 1

Leroy Quet, Oct 09 2000

nonn

Sets of lists of equal size, cf. A000262. - Vladeta Jovovic, Nov 02 2003
From Gus Wiseman, Jan 10 2019: (Start)
Number of matrices whose entries are 1,...,n, up to column permutations. For example, inequivalent representatives of the a(4) = 37 matrices are:
One 1 X 4 matrix:
  [1234]
12 2 X 2 matrices:
  [12] [12] [13] [13] [14] [14] [23] [23] [24] [24] [34] [34]
  [34] [43] [24] [42] [23] [32] [14] [41] [13] [31] [12] [21]
and 24 4 X 1 matrices:
  [1][1][1][1][1][1][2][2][2][2][2][2][3][3][3][3][3][3][4][4][4][4][4][4]
  [2][2][3][3][4][4][1][1][3][3][4][4][1][1][2][2][4][4][1][1][2][2][3][3]
  [3][4][2][4][2][3][3][4][1][4][1][3][2][4][1][4][1][2][2][3][1][3][1][2]
  [4][3][4][2][3][2][4][3][4][1][3][1][4][2][4][1][2][1][3][2][3][1][2][1]
in total 1+12+24 = 37.
(End)

			a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A000005, A005225, A038041, A038048, A061095, A121860, A209903 (exp), A236696, A320444.

Row sums of A066387.

a[n_] := n! DivisorSum[n, 1/#! &]; Array[a, 22] (* Jean-François Alcover, Dec 23 2015 *)

a(n)=n! * sumdiv(n, d, 1/d! );  /* Joerg Arndt, Oct 07 2012 */

E.g.f.: Sum_{n>0} (exp(x^n)-1). - Vladeta Jovovic, Dec 30 2001
E.g.f.: Sum_{k>0} x^k/k!/(1-x^k). - Vladeta Jovovic, Oct 14 2003
Equals the logarithmic derivative of A209903. - Paul D. Hanna, Jul 26 2012

A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

Original entry on oeis.org

1, 2, 3, 13, 25, 301, 721, 10921, 60481, 740881, 3628801, 106777441, 479001601, 12462690241, 134399865601, 2833553923201, 20922789888001, 892191453753601, 6402373705728001, 268633265290790401, 3652732042831872001, 102181898422712908801, 1124000727777607680001

Offset: 1

Vladeta Jovovic, Oct 15 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..450

Cf. A057625, A061095, A038041, A038048, A005225, A209902 (exp).

Array[n \[Function] DivisorSum[n, (n - 1)!/(# - 1)! &], 25] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

a(n)=sumdiv(n,d,(n-1)!/(d-1)!); \\ Joerg Arndt, May 21 2013

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

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A058892 E.g.f.: exp(f(x)-1), where f(x) = o.g.f. for partitions (A000041), Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

1, 1, 5, 31, 265, 2621, 31621, 426595, 6574961, 111673945, 2092318021, 42552808871, 937495160185, 22150499622421, 559765402811525, 15039597200385451, 428293292251548001, 12875707199330296625, 407547173842501629061

Offset: 0

N. J. A. Sloane, Jan 08 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A000041, A038048.

nmax = 30; CoefficientList[Series[1/E*Exp[Product[1/(1 - x^k), {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!(* Vaclav Kotesovec, Aug 19 2015 *)

N=66; q='q+O('q^N);
f=exp( 1/prod(n=1,N, 1-q^n ) - 1 );
egf=serlaplace(f);
Vec(egf)
/* Joerg Arndt, Oct 06 2012 */

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000041(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Oct 15 2017

A008298 Triangle of D'Arcais numbers.

Original entry on oeis.org

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 144, 450, 215, 30, 1, 1440, 3394, 2475, 565, 45, 1, 5760, 30912, 28294, 9345, 1225, 63, 1, 75600, 293292, 340116, 147889, 27720, 2338, 84, 1, 524160, 3032208, 4335596, 2341332, 579369, 69552, 4074, 108, 1, 6531840, 36290736, 57773700, 38049920, 11744775, 1857513, 154350, 6630, 135, 1

Offset: 1

N. J. A. Sloane

Also the Bell transform of A038048(n+1) and the inverse Bell transform of A180563(n+1) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Named after the Italian mathematician Francesco Flores D'Arcais (1849-1927). - Amiram Eldar, Jun 13 2021

			exp(Sum_{n>0} sigma(n)*u*x^n/n) = 1+u*x/1!+(3*u+u^2)*x^2/2!+(8*u+9*u^2+u^3)*x^3/3!+(42*u+59*u^2+18*u^3+u^4)*x^4/4!+...
Triangle starts:
      1:
      3,      1;
      8,      9,      1;
     42,     59,     18,      1;
    144,    450,    215,     30,     1;
   1440,   3394,   2475,    565,    45,    1;
   5760,  30912,  28294,   9345,  1225,   63,  1;
  75600, 293292, 340116, 147889, 27720, 2338, 84, 1;
  ...
T(4; u) = 4!*(binomial(u+3,4) + binomial(u+1,2)*binomial(u,1) + binomial(u+1,2) + binomial(u,1)^2 + binomial(u,1)) = 42*u+59*u^2+18*u^3+u^4.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.
F. D'Arcais, Développement en série, Intermédiaire Math., Vol. 20 (1913), pp. 233-234.

Seiichi Manyama, Rows n = 1..100, flattened (rows n = 1..20 from Vincenzo Librandi)
Peter Luschny, The Bell transform.

Column k=1..3 give A038048, A059356, A059357.
Row sums give A053529.
Cf. A075525, A180563, A210590.

P := proc(n): if n=0 then 1 else P(n):= (1/n)*(add(x(n-k) * P(k), k=0..n-1)) fi; end: with(numtheory): x := proc(n): sigma(n) * x end: Q := proc(n): n!*P(n) end: T := proc(n, k): coeff(Q(n), x, k) end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, Jul 08 2016

t[0][u_] = 1; t[n_][u_] := t[n][u] = Sum[(n-1)!/(n-k)!*DivisorSigma[1, k]*u*t[n-k][u], {k, 1, n}]; row[n_] := CoefficientList[ t[n][u], u] // Rest; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Oct 03 2012, after Vladeta Jovovic *)

row(n)={local(P(n)=if(n,sum(k=0,n-1,sigma(n-k)*x*P(k))/n,1)); Vecrev(P(n)*n!/x)} \\ T(n,k)=row(n)[k]. - M. F. Hasler, Jul 13 2016

a(n) = if(n<1, 0, (n-1)!*sigma(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1))) \\ Seiichi Manyama, Nov 08 2020 after Peter Luschny

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
print(bell_matrix(lambda n: A038048(n+1), 9)) # Peter Luschny, Jan 19 2016

G.f.: Sum_{1<=k<=n} T(n, k)*u^k*t^n/n! = ((1-t)*(1-t^2)*(1-t^3)...)^(-u).
Recurrence for degree n D'Arcais polynomials T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = Sum_{k=1..n} (n-1)!/(n-k)!*sigma(k)*u*T(n-k; u), T(0; u) = 1. - Vladeta Jovovic, Oct 11 2002
T(n; u) = n!*Sum_{pi} Product_{i=1..n} binomial(u+k(i)-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Oct 11 2002
E.g.f.: exp(Sum_{n>0} sigma(n)*u*x^n/n), where sigma(n)=A000203(n). - Vladeta Jovovic, Jan 10 2003
T(n, k) = coeff(n!*P(n), x^k), n >= 1 and 1 <= k <= n, with P(n) = (1/n)*Sum_{k=0..n-1} sigma(n-k)*P(k)*x for n >= 1 and P(n=0) = 1. See A036039. - Johannes W. Meijer, Jul 08 2016
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.

More terms from Vladeta Jovovic, Dec 28 2001

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A318249 a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 4, 18, 48, 480, 1440, 20160, 120960, 1451520, 7257600, 239500800, 958003200, 24908083200, 348713164800, 6538371840000, 41845579776000, 2134124568576000, 12804747411456000, 729870602452992000, 9731608032706560000, 204363768686837760000, 2248001455555215360000, 206816133911079813120000

Offset: 1

Ilya Gutkovskiy, Aug 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..450

Column 1 of A338805.

Cf. A000005, A028342, A038048, A318250, A338814.

Table[(n - 1)! DivisorSigma[0, n], {n, 1, 24}]
nmax = 24; Rest[CoefficientList[Series[Sum[Sum[x^(j k)/(j k), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]

a(n) = (n-1)!*numdiv(n); \\ Michel Marcus, Aug 22 2018

E.g.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^(1/k)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A028342.
a(p^k) = (k + 1)*(p^k - 1)!, where p is a prime.

A265024 a(n) = n! * Sum_{d in D(n+1)} (-1)^(d+1)*(n+1)/d, D(n) the divisors of n.

Original entry on oeis.org

1, 1, 8, 6, 144, 480, 5760, 5040, 524160, 2177280, 43545600, 159667200, 6706022400, 49816166400, 2092278988800, 1307674368000, 376610217984000, 4623936565248000, 128047474114560000, 729870602452992000, 77852864261652480000, 613091306060513280000

Offset: 0

Peter Luschny, Jan 26 2016

nonn

Cf. A000593, A027750, A038048, A075525 (Bell transform).

Rest[CoefficientList[Series[Log[QPochhammer[-1, x]/2], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 15 2017 *)

a(n) = n!*sumdiv(n+1, d, (-1)^(d+1)*(n+1)/d); \\ Michel Marcus, Jan 26 2016

A265024 = lambda n: factorial(n)*sum((-1)^(d+1)*(n+1)/d for d in divisors(n+1))
[A265024(n) for n in (0..21)]

E.g.f.: d/dx log(Product_{k>=1} (1 + x^k)). - Ilya Gutkovskiy, Oct 15 2017
a(n) = n! * A000593(n+1). - Seiichi Manyama, Nov 08 2020.
E.g.f.: d/dx ( Sum_{k>=1} x^k / (k * (1 - x^(2*k))) ). - Seiichi Manyama, Sep 18 2021

A356010 a(n) = n! * Sum_{k=1..n} floor(n/k)/k.

Original entry on oeis.org

1, 5, 23, 134, 814, 6324, 50028, 475824, 4806576, 54597600, 644119200, 8847100800, 121718332800, 1853505158400, 29894856364800, 518855607244800, 9197155541145600, 179420609436364800, 3537039053405491200, 75849875285280768000, 1670700245252548608000

Offset: 1

Seiichi Manyama, Jul 23 2022

nonn

Robert Israel, Table of n, a(n) for n = 1..447

Cf. A038048, A355886.

S:= ListTools:-PartialSums([seq(numtheory:-sigma(k)/k, k=1..30)]):
seq(n! * S[n], n=1..30); # Robert Israel, Aug 10 2025

PARI
```
a(n) = n!*sum(k=1, n, n\k/k);
```

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

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

a(n) = n!*sum(k=1, n, sigma(k)/k); \\ Seiichi Manyama, Aug 03 2022

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k * (1 - x^k)).
E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - x^k).
a(n) ~ c * n! * n, where c = Pi^2/6. - Vaclav Kotesovec, Aug 02 2022
a(n) = n! * Sum_{k=1..n} sigma(k)/k. - Seiichi Manyama, Aug 03 2022

A330353 Expansion of e.g.f. Sum_{k>=1} (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 4, 18, 112, 810, 7144, 73458, 850672, 11069370, 161190904, 2575237698, 44571447232, 836188737930, 16970931765064, 368985732635538, 8524290269083792, 208874053200038490, 5428866923032585624, 149250273758730282978, 4318265042184721248352

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A000041, A000203, A000629, A002745, A008277, A038048, A167137, A308555, A330351, A330352, A330354.

nmax = 20; CoefficientList[Series[Sum[(Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A167137.
G.f.: Sum_{k>=1} (k - 1)! * sigma(k) * x^k / Product_{j=1..k} (1 - j*x), where sigma = A000203.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * sigma(k).
a(n) ~ n! * Pi^2 / (12 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 14 2019

A330354 Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).

Original entry on oeis.org

1, 2, 1, 21, -122, 1752, -21730, 309166, -4521032, 70344768, -1173530712, 21642745704, -448130571696, 10352684535840, -260101132095888, 6921279885508848, -191813249398678272, 5502934340821289088, -163695952380982280832, 5078687529186002247552

Offset: 1

Ilya Gutkovskiy, Dec 11 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A000041, A000203, A002743, A008275, A038048, A089064, A306042, A330351, A330352, A330353, A330494.

nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]

E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A306042.
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of the partition numbers (A000041).
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * sigma(k), where sigma = A000203.
Conjecture: a(n) ~ n! * (-1)^n * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

cp's OEIS Frontend

A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

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A087906 a(n) = Sum_{d|n} (n-1)!/(d-1)!.

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Mathematica

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Extensions

A058892 E.g.f.: exp(f(x)-1), where f(x) = o.g.f. for partitions (A000041), Product_{k>=1} 1/(1-x^k).

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Mathematica

PARI

Formula

A008298 Triangle of D'Arcais numbers.

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Maple

Mathematica

PARI

PARI

Sage

Formula

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A274804 The exponential transform of sigma(n).

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Maple

Mathematica

Formula

A318249 a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).

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Mathematica

PARI

Formula

A265024 a(n) = n! * Sum_{d in D(n+1)} (-1)^(d+1)*(n+1)/d, D(n) the divisors of n.

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