A058892 -id:A058892 - cp's OEIS frontend

A038048 a(n) = (n-1)! * sigma(n).

1, 3, 8, 42, 144, 1440, 5760, 75600, 524160, 6531840, 43545600, 1117670400, 6706022400, 149448499200, 2092278988800, 40537905408000, 376610217984000, 13871809695744000, 128047474114560000, 5109094217170944000

Offset: 1

Christian G. Bower

sigma(n) = A000203(n) is the sum of the divisors of n.
Number of labeled regular octopi (or octopuses, cycles of ordered sets all the same size).
Left edge of triangle in A008298.

			a(6) = 5! * (1 + 2 + 3 + 6) = 1440 = 6! * (1 + 1/2 + 1/3 + 1/6).

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 56 (1.4.67).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159, #10, A(n,1).

T. D. Noe, Table of n, a(n) for n = 1..100
Xiaojun Liu, Motohico Mulase, Adam Sorkin, Quantum curves for simple Hurwitz numbers of an arbitrary base curve, arXiv:1304.0015 [math.AG], 2013.
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.

Cf. A000203, A057625, A058892, A110373, A110374.

a := n -> n!*add(1/j, j=numtheory:-divisors(n)): seq(a(n), n=1..23); # Emeric Deutsch, Jul 24 2005

a[n_] := (n-1)!*DivisorSigma[1, n]; Table[a[n], {n, 20}] (* Jean-François Alcover, Mar 23 2011 *)

a(n)=(n-1)!*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

A038048 = lambda n: factorial(n-1)*sigma(n,1)
[A038048(n) for n in (1..20)] # Peter Luschny, Jan 19 2016

a(n) = Sum_{d|n} n!/d. - Amarnath Murthy, Jul 24 2005
a(p) = (p+1)*(p-1)! if p is a prime. - Amarnath Murthy, Jul 24 2005
E.g.f.: log(f(x)), where f(x) = o.g.f. for partitions (A000041), Product_{k>=1} 1/(1 - x^k). - N. J. A. Sloane
E.g.f.: Sum_{k>0} x^k/(k*(1-x^k)). - Vladeta Jovovic, Mar 27 2005
a(n) = A000142(n-1)*A000203(n). - Omar E. Pol, Feb 26 2014

More terms from Emeric Deutsch, Jul 24 2005

Edited by N. J. A. Sloane, May 12 2008 at the suggestion of Joerg Arndt

A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

Original entry on oeis.org

1, 1, 3, 19, 121, 1041, 10651, 121843, 1575729, 22970881, 366805171, 6365365491, 120044573353, 2430782532049, 52677233993931, 1217023986185491, 29799465317716321, 771272544315151233, 21044341084622337379, 603173026772647474771

Offset: 0

Seiichi Manyama, Oct 17 2017

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 9, 1, 1, 1, 3, 9, 1, ...], a purely periodic sequence with exact period 5.
3 divides a(3*n+2); 9 divides a(9*n+8); 11 divides a(11*n+4); 19 divides a(19*n+3). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A293839, A293841.

Cf. A058892.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k)*a(n-k)/(n-k)! for n > 0.

A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 5, 13, 0, 1, 1, 5, 25, 73, 0, 1, 1, 5, 31, 193, 501, 0, 1, 1, 5, 31, 241, 1601, 4051, 0, 1, 1, 5, 31, 265, 2261, 16741, 37633, 0, 1, 1, 5, 31, 265, 2501, 25501, 190345, 394353, 0, 1, 1, 5, 31, 265, 2621, 29461, 319915, 2509025

Offset: 0

Seiichi Manyama, Oct 25 2017

			Square array B(j,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   1,    2,    2,    2, ...
   0,   1,    2,    3,    3, ...
   0,   1,    3,    4,    5, ...
   0,   1,    3,    5,    6, ...
Square array A(n,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   3,    5,    5,    5, ...
   0,  13,   25,   31,   31, ...
   0,  73,  193,  241,  265, ...
   0, 501, 1601, 2261, 2501, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000262, A294213, A294214, A294215, A294216.
Rows n=0 gives A000012.
Main diagonal gives A058892.
Cf. A058398, A294250, A294254, A294289.

B(j,k) is the coefficient of Product_{i=1..k} 1/(1-x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

Original entry on oeis.org

1, 1, 3, 13, 79, 579, 5209, 53347, 628257, 8223481, 119473291, 1893056781, 32677209103, 606930554923, 12109058077809, 257638964244739, 5830359141736129, 139638723615395697, 3531794326401241747, 93977250969358226701, 2625647922067519041231, 76809884197769914248211

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

Note that exp( Sum_{k>=1} A183610(n,k)*x^k/k ) is an integer series for row n>=1; the partition numbers, which forms row 0 of table A183610, is the exception.

			G.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 579*x^5/5! + 5209*x^6/6! +  ...
such that log(A(x)) = x + 2*x^2/2 + 3*x^3/3 + 5*x^4/4 + 7*x^5/5 + 11*x^6/6 + 15*x^7/7 + 22*x^8/8 + ... + A000041(n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..432

Cf. A183610, A000041, A058892, A293731, A293839.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsP[k]*x^k/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(n):=if n=0 then 1 else (n-1)!*sum(num_partitions(i+1)*a(n-i-1)/(n-i-1)!,i,0,n-1); /* Vladimir Kruchinin, Feb 27 2015 */

{a(n)=n!*polcoeff(exp(sum(m=1,n+1,numbpart(m)*x^m/m+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

a(n) = (n-1)!*sum(p(i+1)*a(n-i-1)/(n-i-1)!,i,0,n-1), a(0)=1, where p(i) is the number of partitions of n. - Vladimir Kruchinin, Feb 27 2015

A300511 Expansion of e.g.f. exp(Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 1, 3, 10, 42, 203, 1119, 6841, 45916, 334414, 2622256, 21984668, 195991611, 1849158088, 18390563792, 192128761836, 2102097270199, 24022460183508, 286060559298908, 3542047217686560, 45517563689858955, 606014811356799054, 8346153294214800894, 118731713512110007282

Offset: 0

Ilya Gutkovskiy, Mar 07 2018

nonn

Exponential transform of A000041.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 42*x^4/4! + 203*x^5/5! + 1119*x^6/6! + 6841*x^7/7! + 45916*x^8/8! + ..

Alois P. Heinz, Table of n, a(n) for n = 0..533
N. J. A. Sloane, Transforms
Index entries for related partition-counting sequences

Cf. A000041, A001970, A058892, A215915, A218481, A300512.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*combinat[numbpart](j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 07 2018

nmax = 23; CoefficientList[Series[Exp[Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[PartitionsP[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A000041(k)*x^k/k!).

A293731 E.g.f.: exp(Sum_{n>=1} nA000041(n)x^n), where A000041(n) is the number of partitions of n.

Original entry on oeis.org

1, 1, 9, 79, 937, 12501, 204361, 3703099, 76460049, 1732292137, 43118784361, 1161659388231, 33771008443129, 1050438417598909, 34839221780655657, 1225699869182970931, 45592202322141065761, 1786608566424333658449, 73553912374465725486409

Offset: 0

Seiichi Manyama, Oct 15 2017

nonn

			a(5) = 4! * (1^2*1*a(4)/4! + 2^2*2*a(3)/3! + 3^2*3*a(2)/2! + 4^2*5*a(1)/1! + 5^2*7*a(0)/0!) = 12501.

Seiichi Manyama, Table of n, a(n) for n = 0..410

Cf. A058892, A215915.

Cf. A000041.

nmax = 20; CoefficientList[Series[E^Sum[k*PartitionsP[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k^2*A000041(k)*a(n-k)/(n-k)! for n > 0.

A294261 E.g.f.: exp(Sum_{n>=1} A081362(n)*x^n).

Original entry on oeis.org

1, -1, 1, -7, 49, -301, 2281, -21211, 260737, -3254329, 41086801, -589336111, 9851907121, -170708882917, 3060177746809, -60544788499651, 1298663388032641, -28777111728560881, 665551703689032097, -16413980708818538839, 428253175770218766001

Offset: 0

Seiichi Manyama, Oct 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..437

Main diagonal of A294289.

Cf. A058892, A293840.

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A081362(k)*a(n-k)/(n-k)! for n > 0.

A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000041(j)x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 9, 31, 79, 1, 1, 17, 79, 265, 579, 1, 1, 33, 211, 937, 2621, 5209, 1, 1, 65, 583, 3433, 12501, 31621, 53347, 1, 1, 129, 1651, 12889, 62141, 204361, 426595, 628257, 1, 1, 257, 4759, 49225, 319461, 1395121, 3703099

Offset: 0

Seiichi Manyama, Oct 16 2017

			Square array begins:
     1,    1,     1,     1,      1, ...
     1,    1,     1,     1,      1, ...
     3,    5,     9,    17,     33, ...
    13,   31,    79,   211,    583, ...
    79,  265,   937,  3433,  12889, ...
   579, 2621, 12501, 62141, 319461, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A215915, A058892, A293731.

Rows n=0-1 give A000012.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j^k*A000041(j)*A(n-j,k)/(n-j)! for n > 0.

A380171 Numerators of coefficients in expansion of exp(-1 + 1 / Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, 1, 5, 31, 265, 2621, 31621, 85319, 6574961, 22334789, 2092318021, 42552808871, 187499032037, 22150499622421, 22390616112461, 15039597200385451, 428293292251548001, 103005657594642373, 407547173842501629061, 2708181047424714819491, 36245898714951203790797

Offset: 0

Ilya Gutkovskiy, Jan 14 2025

			1, 1, 5/2, 31/6, 265/24, 2621/120, 31621/720, 85319/1008, 6574961/40320, ...

Cf. A000041, A017665, A058892, A066186, A067764, A098987, A380271 (denominators).

nmax = 20; CoefficientList[Series[Exp[-1 + 1/Product[1 - x^k, {k, 1, nmax}]], {x, 0, nmax}], x] // Numerator
b[0] = 1; b[n_] := b[n] = (1/n) Sum[k PartitionsP[k] b[n - k], {k, 1, n}]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 20}]

b(0) = 1, b(n) = (1/n) * Sum_{k=1..n} k * A000041(k) * b(n-k), a(n) = numerator of b(n).

A380271 Denominators of coefficients in expansion of exp(-1 + 1 / Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 1008, 40320, 72576, 3628800, 39916800, 95800320, 6227020800, 3487131648, 1307674368000, 20922789888000, 2845499424768, 6402373705728000, 24329020081766400, 187146308321280000, 51090942171709440000, 224800145555521536000, 25852016738884976640000

Offset: 0

Ilya Gutkovskiy, Jan 18 2025

			1, 1, 5/2, 31/6, 265/24, 2621/120, 31621/720, 85319/1008, 6574961/40320, ...

Cf. A000041, A017666, A058892, A066186, A067653, A098988, A380171 (numerators).

nmax = 23; CoefficientList[Series[Exp[-1 + 1/Product[1 - x^k, {k, 1, nmax}]], {x, 0, nmax}], x] // Denominator
b[0] = 1; b[n_] := b[n] = (1/n) Sum[k PartitionsP[k] b[n - k], {k, 1, n}]; a[n_] := Denominator[b[n]]; Table[a[n], {n, 0, 23}]

b(0) = 1, b(n) = (1/n) * Sum_{k=1..n} k * A000041(k) * b(n-k), a(n) = denominator of b(n).

cp's OEIS Frontend

A038048 a(n) = (n-1)! * sigma(n).

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A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

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A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

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A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

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A300511 Expansion of e.g.f. exp(Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

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A293731 E.g.f.: exp(Sum_{n>=1} n*A000041(n)*x^n), where A000041(n) is the number of partitions of n.

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A294261 E.g.f.: exp(Sum_{n>=1} A081362(n)*x^n).

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A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)*A000041(j)*x^j).

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A293731 E.g.f.: exp(Sum_{n>=1} nA000041(n)x^n), where A000041(n) is the number of partitions of n.

A293796 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000041(j)x^j).