A036038 -id:A036038 - cp's OEIS frontend

A008480 Number of ordered prime factorizations of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 3, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2, 2, 4, 1, 12, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1

Offset: 1

Olivier Gérard

a(n) depends only on the prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).
Multinomial coefficients in prime factorization order. - Max Alekseyev, Nov 07 2006
The Dirichlet inverse is given by A080339, negating all but the A080339(1) element in A080339. - R. J. Mathar, Jul 15 2010
Number of (distinct) permutations of the multiset of prime factors. - Joerg Arndt, Feb 17 2015
Number of not divisible chains in the divisor lattice of n. - Peter Luschny, Jun 15 2013

A. Knopfmacher, J. Knopfmacher, and R. Warlimont, "Ordered factorizations for integers and arithmetical semigroups", in Advances in Number Theory, (Proc. 3rd Conf. Canadian Number Theory Assoc., 1991), Clarendon Press, Oxford, 1993, pp. 151-165.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.

T. D. Noe, Table of n, a(n) for n = 1..10000
Steven R. Finch, Kalmar's composition constant, June 5, 2003. [Cached copy, with permission of the author]
Carl-Erik Fröberg, On the prime zeta function, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
Gordon Hamilton's MathPickle, Fractal Multiplication (visual presentation of non-commutative multiplication).
Maxie D. Schmidt, Exact formulas for partial sums of the Möbius function expressed by partial sums weighted by the Liouville lambda function, arXiv:2102.05842 [math.NT], 2021-2022.
Eric Weisstein's World of Mathematics, Multinomial Coefficient.

Cf. A000040, A000142, A000961, A002110, A002033, A050382.
Cf. A036038, A036039, A036040, A080575, A102189.
Cf. A099848, A099849, A112624, A130675.
Cf. A124010, record values and where they occur: A260987, A260633.
Absolute values of A355939.

a008480 n = foldl div (a000142 $ sum es) (map a000142 es)
            where es = a124010_row n
-- Reinhard Zumkeller, Nov 18 2015

a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, May 26 2018

Prepend[ Array[ Multinomial @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
(* Second program: *)
a[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times @@ (ee!)]; Array[a, 101] (* Jean-François Alcover, Sep 15 2019 *)

a(n)={my(sig=factor(n)[,2]); vecsum(sig)!/vecprod(apply(k->k!, sig))} \\ Andrew Howroyd, Nov 17 2018

from math import prod, factorial
from sympy import factorint
def A008480(n): return factorial(sum(f:=factorint(n).values()))//prod(map(factorial,f)) # Chai Wah Wu, Aug 05 2023

def A008480(n):
    S = [s[1] for s in factor(n)]
    return factorial(sum(S)) // prod(factorial(s) for s in S)
[A008480(n) for n in (1..101)]  # Peter Luschny, Jun 15 2013

If n = Product (p_j^k_j) then a(n) = ( Sum (k_j) )! / Product (k_j !).
Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of primes.
a(p^k) = 1 if p is a prime.
a(A002110(n)) = A000142(n) = n!.
a(n) = A050382(A101296(n)). - R. J. Mathar, May 26 2017
a(n) = 1 <=> n in { A000961 }. - Alois P. Heinz, May 26 2018
G.f. A(x) satisfies: A(x) = x + A(x^2) + A(x^3) + A(x^5) + ... + A(x^prime(k)) + ... - Ilya Gutkovskiy, May 10 2019
a(n) = C(k, n) for k = A001222(n) where C(k, n) is defined as the k-fold Dirichlet convolution of A001221(n) with itself, and where C(0, n) is the multiplicative identity with respect to Dirichlet convolution.
The average order of a(n) is asymptotic (up to an absolute constant) to 2A sqrt(2*Pi) log(n) / sqrt(log(log(n))) for some absolute constant A > 0. - Maxie D. Schmidt, May 28 2021
The sums of a(n) for n <= x and k >= 1 such that A001222(n)=k have asymptotic order of the form x*(log(log(x)))^(k+1/2) / ((2k+1) * (k-1)!). - Maxie D. Schmidt, Feb 12 2021
Other DGFs include: (1+P(s))^(-1) in terms of the prime zeta function for Re(s) > 1 where the + version weights the sequence by A008836(n), see the reference by Fröberg on P(s). - Maxie D. Schmidt, Feb 12 2021
The bivariate DGF (1+zP(s))^(-1) has coefficients a(n) / n^s (-1)^(A001221(n)) z^(A001222(n)) for Re(s) > 1 and 0 < |z| < 2 - Maxie D. Schmidt, Feb 12 2021
The distribution of the distinct values of the sequence for n<=x as x->infinity satisfy a CLT-type Erdős-Kac theorem analog proved by M. D. Schmidt, 2021. - Maxie D. Schmidt, Feb 12 2021
a(n) = abs(A355939(n)). - Antti Karttunen and Vaclav Kotesovec, Jul 22 2022
a(n) = A130675(n)/A112624(n). - Amiram Eldar, Mar 08 2024

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

A001710 Order of alternating group A_n, or number of even permutations of n letters.

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000

Offset: 0

N. J. A. Sloane

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003
a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008
First column of A092582. - Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012
a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

			G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Xah Lee, Combinatorics: Loop in n points.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Circular Permutation.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Even Permutation.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Odd Permutation.
Eric Weisstein's World of Mathematics, Transposition Graph.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to groups.
Index to divisibility sequences.

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.

[1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)

PARI
```
{a(n) = if( n<2, n>=0, n!/2)};
```

a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ Paul D. Hanna

A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013

from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023

def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024

;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
;; Antti Karttunen, Dec 19 2015

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010
a(n+1) = (-1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010
a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - Gary Detlefs, May 21 2010
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012
a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) for n>=2. - Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further terms from Simone Severini, Oct 15 2004

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, 98329551, 1191578522, 15543026747, 218668538441, 3285749117475, 52700813279423, 896697825211142, 16160442591627990, 307183340680888755, 6147451460222703502, 129125045333789172825, 2841626597871149750951

Offset: 0

Simon Plouffe

This is the total number of hierarchies of n labeled elements arranged on 1 to n levels. A distribution of elements onto levels is "hierarchical" if a level l+1 contains <= elements than level l. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed. See also A140585. Examples: Let the colon ":" separate two consecutive levels l and l+1. Then n=2 --> 3: {1}{2}, {1}:{2}, {2}:{1}, n=3 --> 10: {1}{2}{3}, {1}{2}:{3}, {3}{1}:{2}, {2}{3}:{1}, {1}:{2}:{3}, {3}:{1}:{2}, {2}:{3}:{1}, {1}:{3}:{2}, {2}:{1}:{3}, {3}:{2}:{1}. - Thomas Wieder, May 17 2008
n identical objects are painted by dipping them into a long row of cans of paint of distinct colors. Begining with the first can and not skipping any cans k, 1<=k<=n, objects are dipped (painted) and not more objects are dipped into any subsequent can than were dipped into the previous can. The painted objects are then linearly ordered. - Geoffrey Critzer, Jun 08 2009
a(n) is the number of partitions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition. a(3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. - Alois P. Heinz, Aug 30 2015
Also the number of ordered set partitions of {1,...,n} with weakly decreasing block sizes. - Gus Wiseman, Sep 03 2018
The parity of a(n) is that of A000110(A000120(n)), so a(n) is even if and only if A000120(n) == 2 (mod 3). - Álvar Ibeas, Aug 11 2020

			For n=3, say the first three cans in the row contain red, white, and blue paint respectively. The objects can be painted r,r,r or r,r,w or r,w,b and then linearly ordered in 1 + 3 + 6 = 10 ways. - _Geoffrey Critzer_, Jun 08 2009
From _Gus Wiseman_, Sep 03 2018: (Start)
The a(3) = 10 ordered set partitions with weakly decreasing block sizes:
  {{1},{2},{3}}
  {{1},{3},{2}}
  {{2},{1},{3}}
  {{2},{3},{1}}
  {{3},{1},{2}}
  {{3},{2},{1}}
  {{2,3},{1}}
  {{1,2},{3}}
  {{1,3},{2}}
  {{1,2,3}}
(End)

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126.
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..450 (first 101 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. E. Hoffman, Updown categories: Generating functions and universal covers, arXiv preprint arXiv:1207.1705 [math.CO], 2012.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, 6 (1999), R2.
S. Schreiber & N. J. A. Sloane, Correspondence, 1980.

Main diagonal of: A226873, A261719, A309973.
Row sums of: A226874, A262071, A327803.
Column k=1 of A309951.
Column k=0 of A327801.
Cf. A000041, A000110, A000258, A000670, A007837, A008277, A008480, A036038, A140585, A178682, A212855, A247551, A300335, A318762.

A005651b := proc(k) add( d/(d!)^(k/d),d=numtheory[divisors](k)) ; end proc:
A005651 := proc(n) option remember; local k ; if n <= 1 then 1; else (n-1)!*add(A005651b(k)*procname(n-k)/(n-k)!, k=1..n) ; end if; end proc:
seq(A005651(k), k=0..10) ; # R. J. Mathar, Jan 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      b(n, i-1) +binomial(n, i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015, Dec 12 2016

Table[Total[n!/Map[Function[n, Apply[Times, n! ]], IntegerPartitions[n]]], {n, 0, 20}] (* Geoffrey Critzer, Jun 08 2009 *)
Table[Total[Apply[Multinomial, IntegerPartitions[n], {1}]], {n, 0, 20}] (* Jean-François Alcover and Olivier Gérard, Sep 11 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[t==1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_] := If[n==0, 1, n!*b[n, 0, n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 20 2015, after Alois P. Heinz *)

a(m,n):=if n=m then 1 else sum(binomial(n,k)*a(k,n-k),k,m,(n/2))+1;
makelist(a(1,n),n,0,17); /* Vladimir Kruchinin, Sep 06 2014 */

a(n)=my(N=n!,s);forpart(x=n,s+=N/prod(i=1,#x,x[i]!));s \\ Charles R Greathouse IV, May 01 2015

{ my(n=25); Vec(serlaplace(prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n)))) } \\ Andrew Howroyd, Dec 20 2017

E.g.f.: 1 / Product (1 - x^k/k!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(-k/d). - Vladeta Jovovic, Oct 14 2002
a(n) ~ c * n!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264... . - Vaclav Kotesovec, May 09 2014
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2 , binomial(n,k)*S(n-k,k))+1, S(n,n)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 06 2014
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A036040 Irregular triangle of multinomial coefficients, read by rows (version 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 10, 15, 60, 15, 20, 45, 15, 1, 1, 7, 21, 35, 21, 105, 70, 105, 35, 210, 105, 35, 105, 21, 1, 1, 8, 28, 56, 35, 28, 168, 280, 210, 280, 56, 420, 280, 840, 105, 70, 560, 420, 56, 210, 28, 1, 1, 9, 36, 84, 126, 36, 252

Offset: 1

N. J. A. Sloane

This is different from A080575 and A178867.
T(n,m) = count of set partitions of n with block lengths given by the m-th partition of n.
From Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occurring in complete Bell polynomials, Faa di Bruno's formula (in its simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (See, e.g., Coffey (2006) and program below.)
The complete Bell polynomial of the first n primes gives A007446. (End)
From Tom Copeland, Apr 29 2011: (Start)
A relation between partition polynomials formed from these "refined" Stirling numbers of the second kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
For simple diagrams of the relation between connected graphs, cumulants, and A036040, see the references on statistical physics below. In some sense, these graphs are duals of the umbral bouquets presented in "Lagrange a la Lah". (End)
These M3 (Abramowitz-Stegun) partition polynomials are the complete Bell polynomials (see a comment above) with recurrence (see the Wikipedia link) B_0 = 1, B_n = Sum_{k=0..n-1} binomial(n-1,k) * B_{n-1-k}*x[k+1], n >= 1. - Wolfdieter Lang, Aug 31 2016
With the indeterminates (x_1, x_2, x_3,...) = (t, -c_2*t, -c_3*t, ...) with c_n > 0, umbrally B(n,a.) =  B(n,t)|{t^n = a_n} = 0 and B(j,a.)B(k,a.) = B(j,t)B(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A134685. - Tom Copeland, Feb 09 2018
For applications to functionals in quantum field theory, see Figueroa et al., Brouder, Kreimer and Yeats, and Balduf. In the last two papers, the Bell polynomials with the indeterminates (x_1, x_2, x_3,...) = (c_1, 2!c_2, 3!c_3, ...) are equivalent to the partition polynomials of A130561 in the indeterminates c_n. - Tom Copeland, Dec 17 2019
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced below. (End)
From Tom Copeland, Jun 12 2021: (Start)
These Bell polynomials and their relations to the Faa di Bruno Hopf bialgebra, correlation functions in quantum field theory, and the moment-cumulant duality are given on pp. 134 -144 of Zeidler.
An interpretation of the coefficients of the polynomials is given in expositions of the exponential formula, or principle, in Cameron et al., Duchamp, Duchamp et al., Labelle and Leroux, and Scott and Sokal along with some history. The simplest applications of this principle are given in A060540. (End)

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  3,  6,  1;
  1,  5, 10, 10, 15, 10,  1;
  1,  6, 15, 10, 15, 60, 15, 20, 45, 15, 1;
  ...
The first partition of 3 (i.e., (3)) induces the set {{1, 2, 3}}, so T(3, 1) = 1; the second one (i.e., (2, 1)) the sets {{1, 2}, {3}}, {{1, 3}, {2}}, and {{2, 3}, {1}}, so T(3, 2) = 3; and the third one (i.e., (1, 1, 1)) the set {{1}, {2}, {3}}, so T(3, 1) = 1. - _Lorenzo Sauras Altuzarra_, Jun 20 2022

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_3".
C. Itzykson and J. Drouffe, Statistical Field Theory Vol. 2, Cambridge Univ. Press, 1989, page 412.
S. Ma, Statistical Mechanics, World Scientific, 1985, page 205.
E. Zeidler, Quantum Field Theory II: Quantum Electrodynamics, Springer, 2009.

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.
F. Brglez, Of n-dimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, Elektrotehniski Vestnik, 78(4): 181-192, 2011.
P. Cameron, C. Krattenthaler, and T. Muller, Decomposable functors and the exponential principle, ll, arXiv:0911.3760 [math.CO], 2011.
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.
Tom Copeland, Lagrange a la Lah, Nov 11, 2011.
Tom Copeland, The creation / raising operators for Appell sequences, Nov 21, 2015.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Dec 21, 2015.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.
G. Duchamp, S. Goodenough, K. Penson, and L. Poinsot, Statistics on graphs, exponential formula, and combinatorial physics, arXiv:0910.0695 [cs.DM], 2010.
K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, arXiv:1409.5664v2 [math.CO], 2015, p. 16. [_Tom Copeland_, Feb 29 2016]
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (p. 41).
P. Guha, Riccati Chain, Higher Order Painlevé Type Equations and Stabilizer Set of Virasoro Orbit, 2006.
D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [math-ph], 2016. [_Tom Copeland_, Nov 23 2016]
G. Labelle and P. Leroux, An extension of the exponential formula in enumerative combinatorics , The Electron. J. Comb., Vol. 3, Issue 2, 1996.
Wolfdieter Lang, First 10 rows and polynomials
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
A. Scott and A. Sokal, Some variants of the exponential formula with applications to the multivariate Tutte polynomials (alias Potts model), arXiv:0803.1477 [math.CO], 2009.
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95. [_Tom Copeland_, Dec 20 2018]
Wikipedia, Bell polynomials

See A080575 for another version.
Row sums are the Bell numbers A000110.
Cf. A036036, A036037, A036038, A036039.
Cf. A000040, A007446, A178866 and A178867 (version 3).
Cf. A130561, A263634, A263916, A134685.
Cf. A127671.
Cf. A060540 for the coefficients of the compositions e^{ x^m/m! }.

with(combinat): nmax:=8: for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036040(n,m):= n!/(mul((t!)^q(t)*q(t)!,t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jun 21 2010, Jul 12 2016

runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[temp=Map[Reverse, Sort@ (Sort/@ IntegerPartitions[w]), {1}]; Apply[Multinomial, temp, {1}]/Apply[Times, (runs/@ temp)!, {1}], {w, 6}]

completeBellMatrix := proc(x,n) // x - vector x[1]...x[m], m>=n
local i,j,M; begin
M := matrix(n,n): // zero-initialized
for i from 1 to n-1 do M[i,i+1] := -1: end_for:
for i from 1 to n do for j from 1 to i do
    M[i,j] := binomial(i-1,j-1)*x[i-j+1]: end_for: end_for:
return (M): end_proc:
completeBellPoly := proc(x, n) begin
return (linalg::det(completeBellMatrix (x,n))): end_proc:
for i from 1 to 10 do print(i, completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

A036040_poly(n,V=vector(n,i,eval(Str('x,i))))={matdet(matrix(n,n,i,j,if(j<=i,binomial(i-1,j-1)*V[n-i+j],-(j==i+1))))} \\ Row n of the sequence is made of the coefficients of the monomials ordered by increasing total order (sum of powers) and then lexicographically. - M. F. Hasler, Nov 16 2013, updated Jul 12 2014

from collections import Counter
def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A036040_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [multinomial(p)//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A036040_row(n))
# Peter Luschny, Dec 18 2016, corrected Apr 30 2022

E.g.f.: A(t) = exp(Sum_{k>=1} x[k]*(t^k)/k!).
T(n,m) is the coefficient of ((t^n)/n!)* x[1]^e(m,1)*x[2]^e(m,2)*...*x[n]^e(m,n) in A(t). Here the m-th partition of n, counted in Abramowitz-Stegun(A-St) order, is [1^e(m,1), 2^e(m,2), ..., n^e(m,n)] with e(m,j) >= 0 and if e(m, j)=0 then j^0 is not recorded.
a(n, m) = n!/Product_{j=1..n} j!^e(m,j)*e(m,j)!, with [1^e(m,1), 2^e(m,2), ..., n^e(m, n)] the m-th partition of n in the mentioned A-St order.
With the notation in the Lang reference, x(1) treated as a variable and D the derivative w.r.t. x(1), a raising operator for the polynomial S(n,x(1)) = P3_n(x[1], ..., x[n]) is R = Sum_{n>=0} x(n+1) D^n / n! ; i.e., R S(n, x(1)) = S(n+1, x(1)). The lowering operator is D; i.e., D S(n, x(1)) = n S(n-1, x(1)). The sequence of polynomials is an Appell sequence, so [S(.,x(1)) + y]^n = S(n, x(1) + y). For x(j) = (-1)^(j-1)* (j-1)! for j > 1, S(n, x(1)) = [x(1) - 1]^n + n [x(1) - 1]^(n-1). - Tom Copeland, Aug 01 2008
Raising and lowering operators are given for the partition polynomials formed from A036040 in the link in "Lagrange a la Lah Part I" on page 22. - Tom Copeland, Sep 18 2011
The n-th row is generated by the determinant of [Sum_{k=0..n-1} (x_(k+1)*(dP_n)^k/k!) - S_n], where dP_n is the n X n submatrix of A132440 and S_n is the n X n submatrix of A129185. The coefficients are flagged by the partitions of n represented by the monomials in the indeterminates x_k. Letting all x_n = t, generates the Bell / Touchard / exponential polynomials of A008277. - Tom Copeland, May 03 2014
The partition polynomials of A036039 are obtained by substituting (n-1)! x[n] for x[n] in the partition polynomials of this entry. - Tom Copeland, Nov 17 2015
-(n-1)! F(n, B(1, x[1]), B(2, x[1], x[2])/2!, ..., B(n, x[1], ..., x[n])/n!) = x[n] extracts the indeterminates of the complete Bell partition polynomials B(n, x[1], ..., x[n]) of this entry, where F(n, x[1], ..., x[n]) are the Faber polynomials of A263916. (Compare with A263634.) - Tom Copeland, Nov 29 2015; Sep 09 2016
T(n, m) = A127671(n, m)/A264753(n, m), n >= 1 and 1 <= m <= A000041(n). - Johannes W. Meijer, Jul 12 2016
From Tom Copeland, Sep 07 2016: (Start)
From the connections among the elementary Schur polynomials and the partition polynomials of A130561, A036039 and this array, the partition polynomials of this array satisfy (d/d(x_m)) P(n, x_1, ..., x_n) = binomial(n,m) * P(n-m, x_1, ..., x_(n-m)) with P(k, x_1, ..., x_n) = 0 for k < 0.
Just as in the discussion and example in A130561, the umbral compositional inverse sequence is given by the sequence P(n, x_1, -x_2, -x_3, ..., -x_n).
(End)
The partition polynomials with an index shift can be generated by (v(x) + d/dx)^n v(x). Cf. Guha, p. 12. - Tom Copeland, Jul 19 2018

More terms from David W. Wilson

Additional comments from Wouter Meeussen, Mar 23 2003

A036039 Irregular triangle of multinomial coefficients of integer partitions read by rows (in Abramowitz and Stegun ordering) giving the coefficients of the cycle index polynomials for the symmetric groups S_n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 40, 90, 120, 15, 40, 45, 15, 1, 720, 840, 504, 420, 504, 630, 280, 210, 210, 420, 105, 70, 105, 21, 1, 5040, 5760, 3360, 2688, 1260, 3360, 4032, 3360, 1260, 1120, 1344, 2520, 1120, 1680, 105, 420, 1120, 420, 112, 210, 28, 1

Offset: 1

N. J. A. Sloane

The sequence of row lengths is A000041(n), n >= 1 (partition numbers).
Number of permutations whose cycle structure is the given partition. Row sums are factorials (A000142). - Franklin T. Adams-Watters, Jan 12 2006
A relation between partition polynomials formed from these "refined" Stirling numbers of the first kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
These cycle index polynomials for the symmetric group S_n are also related to a raising operator / infinitesimal generator for fractional integro-derivatives, involving the digamma function and the Riemann zeta function values at positive integers, and to the characteristic polynomial for the adjacency matrix of complete n-graphs A055137 (cf. MathOverflow link). - Tom Copeland, Nov 03 2012
In the Lang link, replace all x(n) by t to obtain A132393. Furthermore replace x(1) by t and all other x(n) by 1 to obtain A008290. See A274760. - Tom Copeland, Nov 06 2012, Oct 29 2015 - corrected by Johannes W. Meijer, Jul 28 2016
The umbral compositional inverses of these polynomials are formed by negating the indeterminates x(n) for n>1, i.e., P(n,P(.,x(1),-x(2),-x(3),...),x(2),x(3),...) = x(1)^n (cf. A130561 for an example of umbral compositional inversion). The polynomials are an Appell sequence in x(1), i.e., dP(n,x(1))/dx(1) = n P(n-1, x(1)) and (P(.,x)+y)^n=P(n,x+y) umbrally, with P(0,x(1))=1. - Tom Copeland, Nov 14 2014
Regarded as the coefficients of the partition polynomials listed by Lang, a signed version of these polynomials IF(n,b1,b2,...,bn) (n! times polynomial on page 184 of Airault and Bouali) provides an inversion of the Faber polynomials F(n,b1,b2,...,bn) (page 52 of Bouali, A263916, and A115131). For example, F(3, IF(1,b1), IF(2,b1,b2)/2!, IF(3,b1,b2,b3)/3!) = b3 and IF(3, F(1,b1), F(2,b1,b2), F(3,b1,b2,b3))/3! = b3 with F(1,b1) = -b1. (Compare with A263634.) - Tom Copeland, Oct 28 2015; Sep 09 2016
The e.g.f. for the row partition polynomials is Sum_{n>=0} P_n(b_1,...,b_n) x^n/n! = exp[Sum_{n>=1} b_n x^n/n], or, exp[P.(b_1,...,b_n)x] = exp[-], expressed umbrally with <"power series"> denoting umbral evaluation (b.)^n = b_n within the power series. This e.g.f. is central to the paper by Maxim and Schuermannn on characteristic classes (cf. Friedrich and McKay also). - Tom Copeland, Nov 11 2015
The elementary Schur polynomials are given by S(n,x(1),x(2),...,x(n)) = P(n,x(1), 2*x(2),...,n*x(n)) / n!. See p. 12 of Carrell. - Tom Copeland, Feb 06 2016
These partition polynomials are also related to the Casimir invariants associated to quantum density states on p. 3 of Boya and Dixit and pp. 5 and 6 of Byrd and Khaneja. - Tom Copeland, Jul 24 2017
With the indeterminates (x_1,x_2,x_3,...) = (t,-c_2*t,-c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)|{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A133932. - Tom Copeland, Feb 09 2018
For relation to the Witt symmetric functions, as well as the basic power, elementary, and complete symmetric functions, see the Borger link p. 295. For relations to diverse zeta functions, determinants, and paths on graphs, see the MathOverflow question Cycling Through the Zeta Garden. - Tom Copeland, Mar 25 2018
Chmutov et al. identify the partition polynomials of this entry with the one-part Schur polynomials and assert that any linear combination with constant coefficients of these polynomials is a tau function for the KP hierarchy. - Tom Copeland, Apr 05 2018
With the indeterminates in the partition polynomials assigned as generalized harmonic numbers, i.e., as partial sums of the Dirichlet series for the Riemann zeta function, zeta(n), for integer n > 1, sums of simple normalizations of these polynomials give either unity or simple sums of consecutive zeta(n) (cf. Hoffman). Other identities involving these polynomials can be found in the Choi reference in Hoffman's paper. - Tom Copeland, Oct 05 2019
On p. 39 of Ma Luo's thesis is the e.g.f. of rational functions r_n obtained through the (umbral) formula 1/(1-r.T) = exp[log(1+P.T)], a differently signed e.g.f. of this entry, where (P.)^n = P_n are Eisenstein elliptic functions. P. 38 gives the example of 4! * r_4 as the signed 4th row partition polynomial of this entry. This series is equated through a simple proportionality factor to the Zagier Jacobi form on p. 25. Recurrence relations for the P_n are given on p. 24 involving the normalized k-weight Eisenstein series G_k introduced on p. 23 and related to the Bernoulli numbers. - Tom Copeland, Oct 16 2019
The Chern characteristic classes or forms of complex vector bundles and the characteristic polynomials of curvature forms for a smooth manifold can be expressed in terms of this entry's partition polynomials with the associated traces, or power sum polynomials, as the indeterminates. The Chern character is the e.g.f. of these traces and so its coefficients are given by the Faber polynomials with this entry's partition polynomials as the indeterminates. See the Mathoverflow question "A canonical reference for Chern characteristic classes". - Tom Copeland, Nov 04 2019
For an application to the physics of charged fermions in an external field, see Figueroa et al. - Tom Copeland, Dec 05 2019
Konopelchenko, in Proposition 5.2, p. 19, defines an operator P_k that is a differently signed operator version of the partition polynomials of this entry divided by a factorial. These operators give rise to bilinear Hirota equations for the KP hierarchy. These partition polynomials are also presented in Hopf algebras of symmetric functions by Cartier. - Tom Copeland, Dec 18 2019
For relationship of these partition polynomials to calculations of Pontryagin classes and the Riemann xi function, see A231846. - Tom Copeland, May 27 2020
Luest and Skliros summarize on p. 298 many of the properties of the cycle index polynomials given here; and Bianchi and Firrotta, a few on p. 6. - Tom Copeland, Oct 15 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			The partition array T(n, k) begins (see the W. Lang link for rows 1..10):
  n\k   1    2    3    4    5    6    7    8    9   10   11  12   13  14 15 ...
  1:    1
  2:    1    1
  3:    2    3    1
  4:    6    8    3    6    1
  5:   24   30   20   20   15   10    1
  6:  120  144   90   40   90  120   15   40   45   15    1
  7:  720  840  504  420  504  630  280  210  210  420  105  70  105  21  1
... reformatted by _Wolfdieter Lang_, May 25 2019

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2".

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, Pages 179-222.
M. Bianchi and M. Firrotta, DDF operators, open string coherent states and their scattering amplitudes, arXiv:1902.07016 [hep-th], 2019.
J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, Pages 49-70.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
M. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation , arXiv:quant-phy/0302024, 2003.
S. Carrell, Combinatorics of the KP Hierarchy, Thesis, University of Waterloo, Ontario, Canada, 2009.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
S. Chmutov, M. Kazarian, and S. Lando, Polynomial graph invariants and the KP hierarchy , arXiv:1803.09800 [math.CO], p. 16-17, 2018.
Tom Copeland, Lagrange a la Lah, 2011; Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012; The creation / raising operators for Appell sequences, 2015; Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015; Addendum to Elliptic Lie Triad, 2015; Formal group laws and binomial Sheffer polynomials, 2018; Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
Mark Dominus, Cycle classes of permutations [From _Wouter Meeussen_, Jun 26 2009]
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (pp. 40-41).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
M. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values, arXiv:1602.03198 [math.NT], p. 2-3, 2016.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
Wolfdieter Lang, First ten rows and polynomials.
D. Luest and D. Skliros, Handle Operators in String Theory, arXiv:1912.01055 [hep-th], 2019.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, Ph.D. dissertation, Math Dept., Duke Univ., 2018.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, arXiv:1804.06977 [math.NT], 2019.
MathOverflow, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, a question posed by Tom Copeland, 2012.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
L. Maxim and J. Schuermann, Equivariant characteristic classes of external and symmetric products of varieties, arXiv:1508.04356 [math.AG], 2015.
R. Szabo, N=2 gauge theories, instanton moduli spaces and geometric representation theory, arXiv:1507.00685 [hep-th], 2015.
Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.

Cf. A036036, A036037, A036038, A036040.
Cf. other versions based on different partition orderings: A102189 (rows reversed), A181897, A319192.
Cf. A094587, A115131, A130561, A263634, A263916.
Cf. A133932.
Cf. A231846.
Cf. A127671.

nmax:=7: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036039(n, m) := n!/ (mul((t)^q(t)*q(t)!, t=1..n)); od: od: seq(seq(A036039(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016
# 2nd program:
A036039 := proc(n,k)
    local a,prts,e,ai ;
    a := n! ;
    # ASPrts is implemented in A119441
    prts := ASPrts(n)[k] ;
    ai := 1;
    for e from 1 to nops(prts) do
        if e>1 then
            if op(e,prts) = op(e-1,prts) then
                ai := ai+1 ;
            else
                ai := 1;
            end if;
        end if;
        a := a/(op(e,prts)*ai) ;
    end do:
    a ;
end proc:
seq(seq(A036039(n,k),k=1..combinat[numbpart](n)),n=1..15) ; # R. J. Mathar, Dec 18 2016

aspartitions[n_]:=Reverse/@Sort[Sort/@IntegerPartitions[n]];(* Abramowitz & Stegun ordering *);
ascycleclasses[n_Integer]:=n!/(Times@@ #)&/@((#!
Range[n]^#)&/@Function[par,Count[par,# ]&/@Range[n]]/@aspartitions[n])
(* The function "ascycleclasses" is then identical with A&S multinomial M2. *)
Table[ascycleclasses[n], {n, 1, 8}] // Flatten
(* Wouter Meeussen, Jun 26 2009, Jun 27 2009 *)

def PartAS(n):
    P = []
    for k in (1..n):
        Q = [p.to_list() for p in Partitions(n, length=k)]
        for q in Q: q.reverse()
        P = P + sorted(Q)
    return P
def A036039_row(n):
    fn, C = factorial(n), []
    for q in PartAS(n):
        q.reverse()
        p = Partition(q)
        fp = 1; pf = 1
        for a, c in p.to_exp_dict().items():
            fp *= factorial(c)
            pf *= factorial(a)**c
        co = fn//(fp*pf)
        C.append(co*prod([factorial(i-1) for i in p]))
    return C
for n in (1..10):
    print(A036039_row(n)) # Peter Luschny, Dec 18 2016

T(n,k) = n!/Product_{j=1..n} j^a(n,k,j)*a(n,k,j)!, with the k-th partition of  n >= 1 in Abromowitz-Stegun order written as Product_{j=1..n} j^a(n,k,j) with nonnegative integers a(n,k,j) satisfying Sum_{j=1..n} j*a(n,k,j) = n, and the number of parts is Sum_{j=1..n} a(n,k,j) =: m(n,k). - Wolfdieter Lang, May 25 2019
Raising and lowering operators are given for the partition polynomials formed from this sequence in the link in "Lagrange a la Lah Part I" on p. 23. - Tom Copeland, Sep 18 2011
From Szabo p. 34, with b_n = q^n / (1-q^n)^2, the partition polynomials give an expansion of the MacMahon function M(q) = Product_{n>=1} 1/(1-q^n)^n = Sum_{n>=0} PL(n) q^n, the generating function for PL(n) = n! P_n(b_1,...,b_n), the number of plane partitions with sum n. - Tom Copeland, Nov 11 2015
From Tom Copeland, Nov 18 2015: (Start)
The partition polynomials of A036040 are obtained by substituting x[n]/(n-1)! for x[n] in the partition polynomials of this entry.
CIP_n(t-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = P_n(b1,...,bn;t), where CIP_n are the partition polynomials of this entry; F(n,...), those of A263916; and P_n, those defined in my formula in A094587, e.g., P_2(b1,b2;t) = 2 b2 + 2 b1 t + t^2.
CIP_n(-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = n! bn. (End)
From the relation to the elementary Schur polynomials given in A130561 and above, the partition polynomials of this array satisfy (d/d(x_m)) P(n,x_1,...,x_n) = (1/m) * (n!/(n-m)!) * P(n-m,x_1,...,x_(n-m)) with P(k,...) = 0 for k<0. - Tom Copeland, Sep 07 2016
Regarded as Appell polynomials in the indeterminate x(1)=u, the partition polynomials of this entry P_n(u) obey d/du P_n(u) = n * P_{n-1}(u), so the abscissas for the zeros of P_n(u) are the same as those of the extrema of P{n+1}(u). In addition, the coefficient of u^{n-1} in P_{n}(u) is zero since these polynomials are related to the characteristic polynomials of matrices with null main diagonals, and, therefore, the trace is zero, further implying the abscissa for any zero is the negative of the sum of the abscissas of the remaining zeros. This assumes all zeros are distinct and real. - Tom Copeland, Nov 10 2019

More terms from David W. Wilson

Title expanded by Tom Copeland, Oct 15 2020

A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 3, 3, 4, 1, 1, 3, 2, 1, 2, 2, 2, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 4, 3, 5, 1, 1, 4, 2, 1, 3, 3, 1, 3, 2, 2, 4, 1, 1

Offset: 1

N. J. A. Sloane

First differs from A334439 for partitions of 9. Namely, this sequence has (4,4,1) before (5,2,2), while A334439 has (5,2,2) before (4,4,1). - Gus Wiseman, May 08 2020
This is also a list of all the possible prime signatures of a number, arranged in graded colexicographic ordering. - N. J. A. Sloane, Feb 09 2014
This is also the Abramowitz-Stegun ordering of reversed partitions (A036036) if the partitions are reversed again after sorting. Partitions sorted first by sum and then colexicographically are A211992. - Gus Wiseman, May 08 2020

			First five rows are:
{{1}}
{{2}, {1, 1}}
{{3}, {2, 1}, {1, 1, 1}}
{{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
{{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}
Up to the fifth row, this is exactly the same as the reverse lexicographic ordering A080577. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (2,2,2), (3,1,1,1), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)). - _M. F. Hasler_, Jan 23 2020
From _Gus Wiseman_, May 08 2020: (Start)
The sequence of all partitions begins:
  ()         (3,2)        (2,1,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)
  (2)        (2,2,1)      (7)
  (1,1)      (2,1,1,1)    (6,1)
  (3)        (1,1,1,1,1)  (5,2)
  (2,1)      (6)          (4,3)
  (1,1,1)    (5,1)        (5,1,1)
  (4)        (4,2)        (4,2,1)
  (3,1)      (3,3)        (3,3,1)
  (2,2)      (4,1,1)      (3,2,2)
  (2,1,1)    (3,2,1)      (4,1,1,1)
  (1,1,1,1)  (2,2,2)      (3,2,1,1)
  (5)        (3,1,1,1)    (2,2,2,1)
  (4,1)      (2,2,1,1)    (3,1,1,1,1)
(End)

Robert Price, Table of n, a(n) for n = 1..3615, 15 rows.
Wikiversity, Lexicographic and colexicographic order

See A036036 for the graded reflected colexicographic ("Abramowitz and Stegun" or Hindenburg) ordering.
See A080576 for the graded reflected lexicographic ("Maple") ordering.
See A080577 for the graded reverse lexicographic ("Mathematica") ordering: differs from a(48) on!
See A228100 for the Fenner-Loizou (binary tree) ordering.
See also A036038, A036039, A036040: (multinomial coefficients).
Partition lengths are A036043.
Reversing all partitions gives A036036.
The number of distinct parts is A103921.
Taking Heinz numbers gives A185974.
The version ignoring length is A211992.
The version for revlex instead of colex is A334439.
Lexicographically ordered reversed partitions are A026791.
Reverse-lexicographically ordered partitions are A080577.
Sorting partitions by Heinz number gives A296150.
Cf. A000041, A124734, A193073, A228100, A228531, A296774, A334301, A334433, A334436, A334437, A334442.

Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n]],{n,8}] (* Gus Wiseman, May 08 2020 *)
- or -
colen[f_,c_]:=OrderedQ[{Reverse[f],Reverse[c]}];
Join@@Table[Sort[IntegerPartitions[n],colen],{n,8}] (* Gus Wiseman, May 08 2020 *)

Name corrected by Gus Wiseman, May 12 2020

Mathematica programs corrected to reflect offset of one and not zero by Robert Price, Jun 04 2020

A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).
The sequence of row sums is A006128(n).
The number of times k appears in row n is A008284(n,k). - Franklin T. Adams-Watters, Jan 12 2006
The next level (row) gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent. See A128628. - Alford Arnold, Mar 27 2007
The 1's in the (flattened) sequence mark the start of a new row, the value that precedes the 1 equals the row number minus one. (I.e., the 1 preceded by a 0 is the start of row 1, the 1 preceded by a 6 is the start of row 7, etc.) - M. F. Hasler, Jun 06 2018
Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - Gus Wiseman, May 24 2020

			0;
1;
1, 2;
1, 2, 3;
1, 2, 2, 3, 4;
1, 2, 2, 3, 3, 4, 5;
1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7;

Abramowitz and Stegun, Handbook, p. 831, column labeled "m".

T. D. Noe, Rows n = 0..25 of irregular triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 831.
Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.
Wolfdieter Lang, Rows n = 1 ..20.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A036037, A036038, A036039, A036040, A036042.
Row lengths are A000041.
Partition lengths of A036036 and A334301.
The version not sorted by length is A049085.
The generalization to compositions is A124736.
The Heinz number of the same partition is A334433.
The number of distinct elements in the same partition is A334440.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Lexicographically ordered partitions are A193073.
Cf. A103921, A115623, A124734, A185974, A296150, A334435, A334438, A334439.

with(combinat): nmax:=9: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): for k from 1 to y(n) do B(k) := P(n)[k] od: for k from 1 to y(n) do s:=0: j:=0: while sJohannes W. Meijer, Jun 21 2010, revised Nov 29 2012
# alternative implementation based on A119441 by R. J. Mathar, Jul 12 2013
A036043 := proc(n,k)
    local pi;
    pi := ASPrts(n)[k] ;
    nops(pi) ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A036043(n,k)) ;
    end do:
    printf("\n") ;
end do:

Table[Length/@Sort[IntegerPartitions[n]],{n,0,30}] (* Gus Wiseman, May 22 2020 *)

A036043(n,k)=#partitions(n)[k] \\ M. F. Hasler, Jun 06 2018

def A036043_row(n):
    return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (0..10): print(A036043_row(n)) # Peter Luschny, Nov 02 2019

a(n) = A001222(A334433(n)). - Gus Wiseman, May 22 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
a(0) inserted by Franklin T. Adams-Watters, Jun 24 2014
Incorrect formula deleted by M. F. Hasler, Jun 06 2018

cp's OEIS Frontend

A376672 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 5, i.e., numbers m such that A376663(m) = 5.

Views

Author

Keywords

Examples

Links

Crossrefs

A292222 Triangle corresponding to the partition array of the M_1 multinomials (A036038).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A008480 Number of ordered prime factorizations of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

A001710 Order of alternating group A_n, or number of even permutations of n letters.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

SageMath

Scheme

Formula

Extensions

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!*n_2!*...) where (n_1, n_2, ...) runs over all integer partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.