A053529 -id:A053529 - cp's OEIS frontend

A294466 Binomial transform of A053529.

1, 2, 7, 34, 221, 1666, 15187, 153602, 1770169, 22379266, 312164831, 4685997922, 76668261397, 1335425319554, 24921410400811, 493075754663746, 10358312736025457, 228862423291312642, 5335861084579488439, 130235118120543955106, 3333808742649699747661

Offset: 0

Vaclav Kotesovec, Oct 31 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..441

Cf. A218481, A281425, A095051.

Cf. A266232, A294467, A293467, A294468.

Table[Sum[Binomial[n, k]*k!*PartitionsP[k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[x] * x^(1/24)/DedekindEta[Log[x]/(2*Pi*I)], {x, 0, nmax}], x] * Range[0, nmax]!

x='x+O('x^50); Vec(serlaplace(exp(x)/eta(x))) \\ G. C. Greubel, Oct 15 2018

E.g.f.: exp(x)/eta(x), where eta(x) is the Dedekind eta function.
a(n) ~ exp(1) * n! * A000041(n).
a(n) ~ sqrt(2*Pi) * exp(Pi*sqrt(2*n/3) - n + 1) * n^(n - 1/2) / (4*sqrt(3)).
E.g.f.: exp(x + Sum_{k>=1} sigma(k)*x^k/k). - Ilya Gutkovskiy, Oct 15 2018

A364128 Decimal expansion of a constant related to A053529 and A179973.

Original entry on oeis.org

4, 4, 3, 2, 3, 8, 9, 5, 4, 7, 3, 0, 9, 2, 8, 5, 0, 9, 4, 0, 7, 7, 7, 5, 1, 2, 0, 7, 2, 8, 3, 3, 1, 8, 5, 1, 5, 0, 2, 0, 7, 2, 1, 9, 2, 4, 3, 9, 1, 5, 3, 0, 8, 7, 0, 7, 7, 6, 2, 9, 2, 8, 7, 8, 5, 3, 4, 5, 9, 1, 5, 9, 1, 4, 4, 7, 8, 7, 3, 5, 9, 3, 2, 5, 5, 7, 6, 1, 1, 6, 9, 2, 9, 1, 3, 8, 2, 8, 7, 1, 6, 4, 8, 5, 8, 8

Offset: 0

Alois P. Heinz, Jul 09 2023

The digits of this constant are also the final digits of A179973(n) and of the n-th partial sum of A053529 for n -> oo, read from right.

			0.443238954730928509407775120728331851502072192439153087... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A004086, A025016, A053529, A055642, A179973.

Equals lim_{n->oo} A004086(A179973(n))/10^A055642(A179973(n)).

Equals lim_{n->oo} A004086(s(n))/10^A055642(s(n)) with s(n) = Sum_{j=0..n} A053529(j).

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576, 532985208200576, 3020676745975552

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) is also the number of n X n symmetric permutation matrices.
a(n) is also the number of matchings (Hosoya index) in the complete graph K(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - Eric W. Weisstein, May 22 2017
Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - Don Knuth, Mar 31 2008
a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n. - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - Karol A. Penson, Apr 22 2003
Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n). - Roberto E. Martinez II, Jan 09 2002
The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, ... (A001147 with interpolated zeros). - Philippe Deléham, Jun 10 2005
Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry, Jan 12 2006
a(n) is the number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - David Callan, Mar 07 2006
Equals row sums of triangle A152736 starting with offset 1. - Gary W. Adamson, Dec 12 2008
Proof of the recurrence relation a(n) = a(n-1) + (n-1)*a(n-2): number of involutions of [n] containing n as a fixed point is a(n-1); number of involutions of [n] containing n in some cycle (j, n), where 1 <= j <= n-1, is (n-1) times the number of involutions of [n] containing the cycle (n-1 n) = (n-1)*a(n-2). - Emeric Deutsch, Jun 08 2009
Number of ballot sequences (or lattice permutations) of length n. A ballot sequence B is a string such that, for all prefixes P of B, h(i) >= h(j) for i < j, where h(x) is the number of times x appears in P. For example, the ballot sequences of length 4 are 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1231, and 1234. The string 1221 does not appear in the list because in the 3-prefix 122 there are two 2's but only one 1. (Cf. p. 176 of Bruce E. Sagan: "The Symmetric Group"). - Joerg Arndt, Jun 28 2009
Number of standard Young tableaux of size n; the ballot sequences are obtained as a length-n vector v where v_k is the (number of the) row in which the number r occurs in the tableaux. - Joerg Arndt, Jul 29 2012
Number of factorial numbers of length n-1 with no adjacent nonzero digits. For example the 10 such numbers (in rising factorial radix) of length 3 are 000, 001, 002, 003, 010, 020, 100, 101, 102, and 103. - Joerg Arndt, Nov 11 2012
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013
a(n) is the number of permutations of [n] that avoid the consecutive patterns 123 and 132. Proof. Write a self-inverse permutation in standard cycle form: smallest entry in each cycle in first position, first entries decreasing. For example, (6,7)(3,4)(2)(1,5) is in standard cycle form. Then erase parentheses. This is a bijection to the permutations that avoid consecutive 123 and 132 patterns. - David Callan, Aug 27 2014
Getu (1991) says these numbers are also known as "telephone numbers". - N. J. A. Sloane, Nov 23 2015
a(n) is the number of elements x in the symmetric group S_n such that x^2 = e where e is the identity. - Jianing Song, Aug 22 2018 [Edited on Jul 24 2025]
a(n) is the number of congruence orbits of upper-triangular n X n matrices on skew-symmetric matrices, or the number of Borel orbits in largest sect of the type DIII symmetric space SO_{2n}(C)/GL_n(C). Involutions can also be thought of as fixed-point-free partial involutions. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020
From Thomas Anton, Apr 20 2020: (Start)
Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
Apparently a(n) = 2^(f(n mod 4)+floor(n/4))*q where f:{0,1,2,3}->{0,1,2} is given by f(0),f(1)=0, f(2)=1 and f(3)=2 and q is odd. (End)
From Iosif Pinelis, Mar 12 2021: (Start)
a(n) is the n-th initial moment of the normal distribution with mean 1 and variance 1. This follows because the moment generating function of that distribution is the e.g.f. of the sequence of the a(n)'s.
The recurrence a(n) = a(n-1) + (n-1)*a(n-2) also follows, by writing E(Z+1)^n=EZ(Z+1)^(n-1)+E(Z+1)^(n-1), where Z is a standard normal random variable, and then taking the first of the latter two integrals by parts. (End)

			Sequence starts 1, 1, 2, 4, 10, ... because possibilities are {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA}. - _Henry Bottomley_, Jan 16 2001
G.f. = 1 + x + 2*x^2 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 764*x^9 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(4) = 10 standard Young tableaux:
  1 2 3 4
.
  1 2   1 3   1 2 3   1 2 4   1 3 4
  3 4   2 4   4       3       2
.
  1 2   1 3   1 4
  3     2     2
  4     4     3
.
  1
  2
  3
  4
The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
             {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                        {{1,3},{2}}    {{1,4},{2,3}}
                        {{1},{2},{3}}  {{1},{2},{3,4}}
                                       {{1},{2,3},{4}}
                                       {{1,2},{3},{4}}
                                       {{1},{2,4},{3}}
                                       {{1,3},{2},{4}}
                                       {{1,4},{2},{3}}
                                       {{1},{2},{3},{4}}
(End)

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Index entries for "core" sequences
Index entries for sequences related to Young tableaux.
Index entries for related partition-counting sequences

Cf. A001147, A001470, A053529, A099174, A136281-A136283, A001680, A001681.
See also A005425 for another version of the switchboard problem.
Equals 2 * A001475(n-1) for n>1.
First column of array A099020.
A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
Cf. A152736, A128229. - Gary W. Adamson, Dec 12 2008
Diagonal of A182172. - Alois P. Heinz, May 30 2012
Cf. A001813, A006882, A135401, A297708. - Manfred Scheucher, Jan 07 2018
Row sums of: A047884, A049403, A096713 (absolute value), A100861, A104556 (absolute value), A111924, A117506 (M_4 numbers), A122848, A238123.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A000700, A000701, A321729, A321730, A321737.

a000085 n = a000085_list !! n
  a000085_list = 1 : 1 : zipWith (+)
    (zipWith (*) [1..] a000085_list) (tail a000085_list) -- Reinhard Zumkeller, May 16 2013

A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else procname(n-1)+(n-1)*procname(n-2); fi; end;
with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, labeled]:seq(count(ZL3,size=n),n=0..25); # Zerinvary Lajos, Sep 24 2007
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(2):seq(count(A, size=n), n=0..25); # Zerinvary Lajos, Jun 11 2008

<Roger L. Bagula, Oct 06 2006 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2013 *)
a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* Michael Somos, Jun 01 2013 *)
Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* Emanuele Munarini, Mar 02 2016 *)
a[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 18 2017, after Emanuele Munarini *)
RecurrenceTable[{a[n] == a[n-1] + (n-1)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 20}] (* Joan Ludevid, Jun 17 2022 *)
sds[{}]:={{}};sds[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sds[Complement[set,s]]]/@Cases[Subsets[set,{1,2}],{i,_}]; Table[Length[sds[Range[n]]],{n,0,10}] (* Gus Wiseman, Jan 11 2021 *)

B(n,x):=sum(stirling2(n,k)*x^k,k,0,n);
  a(n):=sum(stirling1(n,k)*2^k*B(k,1/2),k,0,n);
  makelist(a(n),n,0,40); /* Emanuele Munarini, May 16 2014 */

makelist((%i/sqrt(2))^n*hermite(n,-%i/sqrt(2)),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* Michael Somos, Nov 15 2002 */

N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ Joerg Arndt, Mar 07 2013

from math import factorial
def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<>1)+1)) # Chai Wah Wu, Aug 31 2023

A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
[simplify(A000085(n)) for n in range(28)] # Peter Luschny, Aug 21 2014

def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
for n in range(100): print(n, a85(n)) # Manfred Scheucher, Jan 07 2018

D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
E.g.f.: exp(x+x^2/2).
a(n) = a(n-1) + A013989(n-2) = A013989(n)/(n+1) = 1+A001189(n).
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - Philippe Deléham, Mar 05 2004
For n>1, a(n) = 2*(A000900(n) + A000902(floor(n/2))). - Max Alekseyev, Oct 31 2015
The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - Karol A. Penson, May 16 2002
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
For asymptotics see the Robinson paper.
a(n) = Sum_{m=0..n} A099174(n,m). - Roger L. Bagula, Oct 06 2006
O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
From Gary W. Adamson, Dec 29 2008: (Start)
a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
Starting with offset 1 = eigensequence of triangle A128229. (End)
a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - Groux Roland, Mar 14 2011
Row sums of |A096713|. a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A047974 and A080599. - Peter Bala, Dec 07 2011
From Sergei N. Gladkovskii, Dec 03 2011 - Oct 28 2013: (Start)
Continued fractions:
E.g.f.: 1+x*(2+x)/(2*G(0)-x*(2+x)) where G(k)=1+x*(x+2)/(2+2*(k+1)/G(k+1)).
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
G.f.: T(0)/(1-x) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)^2/T(k+1)). (End)
a(n) ~ (1/sqrt(2)) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014
a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*2^k*B(k,1/2), where the s(n,k) are (signless) Stirling numbers of the first kind, and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials, where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 16 2014
a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - Peter Luschny, Aug 21 2014
0 = a(n)*(+a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(-a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Aug 22 2014
From Peter Bala, Oct 06 2021: (Start)
a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
Hence for each odd k, the sequence obtained by taking a(n) modulo k is a periodic sequence and the exact period divides k. For example, taking a(n) modulo 7 gives the purely periodic sequence [1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, ...] of period 7. For similar results see A047974 and A115329. (End)

A061256 Euler transform of sigma(n), cf. A000203.

Original entry on oeis.org

1, 1, 4, 8, 21, 39, 92, 170, 360, 667, 1316, 2393, 4541, 8100, 14824, 26071, 46422, 80314, 139978, 238641, 408201, 686799, 1156062, 1920992, 3189144, 5238848, 8589850, 13963467, 22641585, 36447544, 58507590, 93334008, 148449417, 234829969, 370345918

Offset: 0

Vladeta Jovovic, Apr 21 2001

This is also the number of ordered triples of permutations f, g, h in Symm(n) which all commute, divided by n!. This was conjectured by Franklin T. Adams-Watters, Jan 16 2006, and proved by J. R. Britnell in 2012.
According to a message on a blog page by "Allan" (see Secret Blogging Seminar link) it appears that a(n) = number of conjugacy classes of commutative ordered pairs in Symm(n).
John McKay (email to N. J. A. Sloane, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - N. J. A. Sloane, May 19 2013

			1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 92*x^6 + 170*x^7 + 360*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
J. R. Britnell, A formal identity involving commuting triples of permutations, arXiv:1203.5079 [math.CO], 2012.
J. R. Britnell, A formal identity involving commuting triples of permutations, Preprint 2012. - _N. J. A. Sloane_, Jun 13 2012
J. R. Britnell, A formal identity involving commuting triples of permutations, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013.
E. Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, arXiv preprint arXiv:1202.1311 [math.RT], 2012. - _N. J. A. Sloane_, Jun 10 2012
Secret Blogging Seminar, A peculiar numerical coincidence.
N. J. A. Sloane, Transforms
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), this sequence (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A000203, A001001, A001970, A053529, A061255, A061257, A006908, A192065.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 18 2012 *)
nmax = 40; CoefficientList[Series[Product[1/QPochhammer[x^k]^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)^j); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))^2/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

a(n) = A072169(n) / n!.
G.f.: Product_{k=1..infinity} (1 - x^k)^(-sigma(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*sigma(d), cf. A001001.
G.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)^2 /n ). [Paul D. Hanna, Mar 28 2009]
G.f.: exp( Sum_{n>=1} sigma_2(n)*x^n/(1-x^n)/n ). [Vladeta Jovovic, Mar 28 2009]
G.f.: prod(n>=1, E(x^n)^n ) where E(x) = prod(k>=1, 1-x^k). [Joerg Arndt, Apr 12 2013]
a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2 - Pi^(4/3) * n^(1/3) / (4 * 3^(2/3) * Zeta(3)^(1/3)) - 1/24 - Pi^2/(288*Zeta(3))) * A^(1/2) * Zeta(3)^(11/72) / (2^(11/24) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018

Entry revised by N. J. A. Sloane, Jun 13 2012

A181162 Number of commuting functions: the number of ordered pairs (f,g) of functions from {1..n} to itself such that fg=gf (i.e., f(g(i))=g(f(i)) for all i).

Original entry on oeis.org

1, 1, 10, 141, 2824, 71565, 2244096, 83982199, 3681265792, 186047433225, 10716241342240, 697053065658411, 50827694884298784, 4129325095108122637, 371782656333674104624, 36918345387693628911375, 4025196918605160943576576, 479796375191949916361466897

Offset: 0

Jeffrey Norden, Oct 07 2010

Also, the total number of endomorphisms of all directed graphs on n labeled vertices with outdegree of each vertex equal 1. - Max Alekseyev, Jan 09 2015
Seems to be relatively hard to compute for large n. (a(n)-n^n)/2 is always an integer, since it gives the number of unordered pairs of distinct commuting functions.
a(n) is divisible by n as proved by Holloway and Shattuck (2015).
From Joerg Arndt, Jul 21 2014: (Start)
Multiply fg=gf from the right by f to obtain fgf=gff, and use f(gf)=f(fg)=ffg to see ffg=gff; iterate to see f^k g = g f^k for all k>=1; by symmetry g^k f = f g^k holds as well.
More generally, if X and Y are words of length w over the alphabet {f,g}, then X = Y (as functional composition) whenever both words contain j symbols f and k symbols g (and j+k=w).  (End)
Functions with the same mapping pattern have the same number of commuting functions, so there is no need to check every pair. - Martin Fuller, Feb 01 2015

			The a(2) = 10 pairs of maps [2] -> [2] are:
01:  [ 1 1 ]  [ 1 1 ]
02:  [ 1 1 ]  [ 1 2 ]
03:  [ 1 2 ]  [ 1 1 ]
04:  [ 1 2 ]  [ 1 2 ]
05:  [ 1 2 ]  [ 2 1 ]
06:  [ 1 2 ]  [ 2 2 ]
07:  [ 2 1 ]  [ 1 2 ]
08:  [ 2 1 ]  [ 2 1 ]
09:  [ 2 2 ]  [ 1 2 ]
10:  [ 2 2 ]  [ 2 2 ]
- _Joerg Arndt_, Jul 22 2014

Martin Fuller, Table of n, a(n) for n = 0..20
Joerg Arndt, the a(3) = 141 pairs of maps [3] -> [3]
Stephen M. Buckley, Minimal order semigroups with specified commuting probability, 04-03-2013. - _W. Edwin Clark_, Jul 21 2014
Martin Fuller, a(6) from the A001372(6)=130 mapping patterns
M. Holloway and M. Shattuck, Commuting pairs of functions on a finite set, 2015.
Math Overflow, What is the probability two random maps on n symbols commute?, 2013. - _W. Edwin Clark_, Jul 21 2014
Math Overflow, Counting and understanding commuting functions, 2010.

A053529 is a similar count for permutations. A254529 is for permutations commuting with functions.

Cf. A000312, A001372, A239749-A239785, A239836-A239841.

(* This brute force code allows to get a few terms *)
a[n_] := a[n] = If[n == 0, 1, Module[{f, g, T}, T = Tuples[Range[n], n]; Table[f = T[[j, #]]&; g = T[[k, #]] &; Table[True, {n}] == Table[f[g[i]] == g[f[i]], {i, n}], {j, n^n}, {k, n^n}] // Flatten // Count[#, True]&]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 5}] (* Jean-François Alcover, Sep 24 2022 *)

a(11)-a(20) from Martin Fuller, Feb 01 2015

A003293 Number of planar partitions of n decreasing across rows.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 56, 90, 143, 223, 348, 532, 811, 1224, 1834, 2725, 4031, 5914, 8638, 12540, 18116, 26035, 37262, 53070, 75292, 106377, 149738, 209980, 293473, 408734, 567484, 785409, 1083817, 1491247, 2046233, 2800125, 3821959, 5203515

Offset: 0

N. J. A. Sloane

Also number of planar partitions monotonically decreasing down antidiagonals (i.e., with b(n,k) <= b(n-1,k+1)). Transpose (to get planar partitions decreasing down columns), then take the conjugate of each row. - Franklin T. Adams-Watters, May 15 2006
Also number of partitions into one kind of 1's and 2's, two kinds of 3's and 4's, three kinds of 5's and 6's, etc. - Joerg Arndt, May 01 2013
Also count of semistandard Young tableaux with sum of entries equal to n (row sums of A228125). - Wouter Meeussen, Aug 11 2013

			From _Gus Wiseman_, Jan 17 2019: (Start)
The a(6) = 21 plane partitions with strictly decreasing columns (the count is the same as for strictly decreasing rows):
  6   51   42   411   33   321   3111   222   2211   21111   111111
.
  5   4   41   31   32   311   22   221   2111
  1   2   1    2    1    1     11   1     1
.
  3
  2
  1
(End)

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 133.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Nathaniel Johnston)
M. S. Cheema and W. E. Conway, Numerical investigation of certain asymptotic results in the theory of partitions, Math. Comp., 26 (1972), 999-1005.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
B. Gordon and L. Houten, Notes on Plane Partitions I, J. of Comb. Theory, 4 (1968), 72-80.
B. Gordon and L. Houten, Notes on Plane Partitions II, J. of Comb. Theory, 4 (1968), 81-99.
Basil Gordon and Lorne Houten, Notes on plane partitions III (first page is available), Duke Math. J. Volume 36, Number 4 (1969), 801-824.
B. Gordon and L. Houten, Notes on Plane Partitions V, Journal of Combinatorial Theory, vol. 11, issue 2, 1971, pp. 157-168.
B. Gordon and L. Houten, Notes on Plane Partitions VI, Discrete Mathematics, vol. 26, issue 1, 1979, pp. 41-45.
Vaclav Kotesovec, Graph - asymptotic ratio for 10000 terms.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Richard P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279.
Richard P. Stanley, Theory and Application of Plane Partitions. Part 2, Studies in Appl. Math., 1 (1971), 259-279.

Cf. A005308, A005986.

Cf. A000085, A000219, A053529, A138178, A323432, A323436.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(modp(n,2)=0,n,n+1)/2): seq(a(n), n=0..45);  # Alois P. Heinz, Sep 08 2008

CoefficientList[Series[Product[(1-x^k)^(-Ceiling[k/2]), {k, 1, 40}], {x, 0, 40}], x][[1 ;; 40]] (* Jean-François Alcover, Apr 18 2011, after Michael Somos *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^((2*k+1-(-1)^k)/4),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)
nmax = 50; CoefficientList[Series[Product[1/((1-x^(2*k-1))*(1-x^(2*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2015 *)

{a(n)=if(n<0, 0, polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n))} /* Michael Somos, Sep 19 2006 */

G.f.: Product_(1 - x^k)^{-c(k)}, c(k) = 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ....
Euler transform of A110654. - Michael Somos, Sep 19 2006
a(n) ~ 2^(-3/4) * (3*Pi*Zeta(3))^(-1/2) * (n/Zeta(3))^(-49/72) * exp(3/2*Zeta(3) * (n/Zeta(3))^(2/3) + Pi^2*(n/Zeta(3))^(1/3)/24 - Pi^4/(3456*Zeta(3)) + Zeta'(-1)/2) [Basil Gordon and Lorne Houten, 1969]. - Vaclav Kotesovec, Feb 28 2015

More terms from James Sellers, Feb 06 2000

Additional comments from Michael Somos, May 19 2000

A088311 Number of sets of lists with distinct list sizes, cf. A000262.

Original entry on oeis.org

1, 1, 2, 12, 48, 360, 2880, 25200, 241920, 2903040, 36288000, 479001600, 7185024000, 112086374400, 1917922406400, 35307207936000, 669529276416000, 13516122267648000, 294509190463488000, 6568835422076928000, 155705728523304960000, 3882911605049917440000

Offset: 0

Vladeta Jovovic, Nov 05 2003

nonn

a(n) also enumerates ordered pairs of permutation functions on n elements where f(g(x)) = g(g(f(x))). - Chad Brewbaker, Mar 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000009, A007837, A007838.

Other ordered permutation function pair relations are A000012, A000085, A000142, A001044, A053529.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( (&*[1+x^j: j in [1..m+2]]) ))); // G. C. Greubel, Dec 14 2022

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> n!*b(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nn = 19; Drop[ Range[0, nn]! CoefficientList[ Series[ Product[1 + x^i, {i,nn}], {x,0,nn}], x], 0] (* Geoffrey Critzer, Aug 05 2013; adapted to new offset by Vincenzo Librandi, Mar 28 2014 *)
nmax = 20; CoefficientList[Series[Product[1/(1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 19 2015 *)

my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ Joerg Arndt, Aug 06 2013

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1) #  Peter Luschny's code of A000009 and A166861
b = EulerTransform(a)
[factorial(n)*b(n) for n in range(41)] # G. C. Greubel, Dec 14 2022

E.g.f: Product_{m>0} (1+x^m).

a(n) = n! * A000009(n).

Prepended a(0) = 1, Joerg Arndt, Aug 06 2013

A101509 Binomial transform of tau(n) (see A000005).

Original entry on oeis.org

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773

Offset: 0

Paul Barry, Dec 05 2004

Row sums of A101508.

Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011

a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)

A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

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).

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