A036038 -id:A036038 - cp's OEIS frontend

A035206 Number of multisets associated with least integer of each prime signature.

1, 1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 105, 140, 420, 140, 105, 210, 42, 1, 8, 56, 56, 56, 28, 168, 336, 336, 168, 168, 280, 840, 420, 840, 70, 280, 1120, 560, 168, 420, 56, 1, 9, 72

Offset: 0

Alford Arnold

a(n,k) multiplied by A036038(n,k) yields A049009(n,k).
a(n,k) enumerates distributions of n identical objects (balls) into m of altogether n distinguishable boxes. The k-th partition of n, taken in the Abramowitz-Stegun (A-St) order, specifies the occupation of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of parts of the k-th partition of n. For the A-St ordering see pp.831-2 of the reference given in A117506. - Wolfdieter Lang, Nov 13 2007
The sequence of row lengths is p(n)= A000041(n) (partition numbers).
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
The corresponding triangle with summed row entries which belong to partitions of the same number of parts k is A103371. [Wolfdieter Lang, Jul 11 2012]

			n\k 1  2  3  4   5   6   7   8   9  10  11  12  13 14 15
0   1
1   1
2   2  1
3   3  6  1
4   4 12  6 12   1
5   5 20 20 30  30  20   1
6   6 30 30 15  60 120  20  60  90  30   1
7   7 42 42 42 105 210 105 105 140 420 140 105 210 42  1
...
Row No. 8:  8  56 56 56 28 168 336 336 168 168 280  840 420 840 70 280 1120 560 168 420 56 1
Row No. 9: 9 72 72 72 72 252 504 504 252 252 504 84 504 1512 1512 1512 1512 504 630 2520 1260 3780 630 504 2520 1680 252 756 72 1
[rewritten and extended table by _Wolfdieter Lang_, Jul 11 2012]
a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable (identical) balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. _Wolfdieter Lang_, Nov 13 2007

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
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].
Wolfdieter Lang, First 10 rows and more.

Cf. A036038, A048996, A049009.
Cf. A001700 (row sums).
Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).

C(sig)={my(S=Set(sig)); binomial(vecsum(sig), #sig)*(#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n) and m(n,k):=A036043(n,k) gives the number of parts of the k-th partition of n.

More terms from Joshua Zucker, Jul 27 2006

a(0)=1 prepended by Andrew Howroyd, Oct 18 2020

A212359 Partition array for the number of representative necklaces (only cyclic symmetry) with n beads, each available in n colors. Only the color type (signature) matters.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 6, 1, 1, 2, 4, 6, 12, 24, 1, 1, 3, 4, 5, 10, 16, 20, 30, 60, 120, 1, 1, 3, 5, 6, 15, 20, 30, 30, 60, 90, 120, 180, 360, 720, 1, 1, 4, 7, 10, 7, 21, 35, 54, 70, 42, 105, 140, 210, 318, 210, 420, 630, 840, 1260, 2520, 5040

Offset: 1

Wolfdieter Lang, Jun 25 2012

The row lengths sequence is A000041(n), n>=1.
The partitions are ordered like in Abramowitz-Stegun (A-St order). For the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used.
A necklace with n beads (n-necklace) has here only the cyclic C_n symmetry group. This is in contrast to, e.g., the Harary-Palmer reference, p. 44, where a n-necklace has the symmetry group D_n, the dihedral group of degree n (order 2n), which allows, in addition to C_n operations, also a turnover (in 3-space) or a reflection (in 2-space).
The necklace number a(n,k) gives the number of n-necklaces, with up to n colors for each bead, belonging to the k-th partition of n in A-St order in the following way. Write this partition with nonincreasing parts (this is the reverse of the partition as given by A-St), e.g., [3,1^2], not [1^2,3], is written as [3,1,1], a partition of n=5. In general [p[1],p[2],...,p[m]], with p[1]>=p[2]>=...>=p[m]>=1, with m the number of parts. To each such partition of n corresponds an n-multiset obtained by 'exponentiation'. For the given example the 5-multiset is {1^3,2^1,3^1}={1,1,1,2,3}. In general {1^p[1],2^p[2],...,m^p[m]}. Such an n-multiset representative (of a repetition class defined by the exponents, sometimes called signature) encodes the n-necklace color monomial by c[1]^p[1]*c[2]^p[2]*...*c[m]^p[m]. For the example one has c[1]^3*c[2]*c[3]. The number of 5-necklaces with this color assignment is a(5,4) because [3,1,1] is the 4th partition of 5 in A-St order. The a(5,4)=4 non-equivalent 5-necklaces with this color assignment are cyclic(c[1]c[1]c[1]c[2]c[3]), cyclic(c[1]c[1]c[1]c[3]c[2]), cyclic(c[1]c[1]c[2]c[1]c[3]) and cyclic(c[1]c[1]c[3]c[1]c[2]).
Such a set of a(n,k) n-necklaces for the given color assignment stands for other sets of the same order where different colors from the repertoire {c[1],...,c[n]} are chosen. In the example, the partition [3,1,1]  with the representative multiset [1^3,2,3] stands for all-together 5*binomial(4,2) = 30 such sets, each leading to 4 possible non-equivalent 5-necklace arrangements. Thus one has, in total, 30*4=120 5-necklaces with color signature determined from the partition [3,1,1]. See the partition array A212360 for these numbers.
For the example n=4, k=1..5, see the Stanley reference, last line, where the numbers a(4,k) are, in A-St order, 1, 1, 2, 3, 6, summing to A072605(4).
a(n,k) is computed from the cycle index Z(C_n) for the cyclic group (see A212357 and the link given there) after the variables x_j have been replaced by the j-th power sum sum(c[i]^j,i=1..n), abbreviated as Z(C_n,c_n) with c_n:=sum(c[i],i=1..n), n>=1. The coefficient of the color assignment representative determined by the k-th partition of n in A-St order, as explained above, is a(n,k). See the Harary-Palmer reference, p. 36, Theorem (PET) with A=C_n and p. 36 eq. (2.2.10) for the cycle index polynomial Z(C_n). See the W. Lang link for more details.
The corresponding triangle with summed entries of row n which belong to partitions of n with the same number of parts is A213934. [Wolfdieter Lang, Jul 12 2012]

			n\k  1 2 3 4 5  6  7  8  9 10  11  12  13  14  15
1    1
2    1 1
3    1 1 2
4    1 1 2 3 6
5    1 1 2 4 6 12 24
6    1 1 3 4 5 10 16 20 30 60 120
7    1 1 3 5 6 15 20 30 30 60  90 120 180 360 720
...
See the link for the rows n=1..15 and the corresponding color polynomials for n=1..10.
a(4,5)=6 because the partition in question is 1^4, the corresponding color type representative multinomial is c[1]*c[2]*c[3]*c[4] (all four colors are involved), and there are the 6 C_4 non-equivalent 4-necklaces (we use here j for color c[j]): 1234, 1243, 1324, 1342, 1423 and 1432 (all taken as cyclically). For this partition there is only one color choice.
a(4,4)=3 because the partition is [2,1^2]=[2,1,1], the color representative monomial is c[1]^2*c[2]*c[3], and the arrangements are 1123, 1132  and  1213 (all taken cyclically). There are, in total, 4*binomial(3,2)=12 color multinomials of this signature (color type) in Z(C_4,c_4).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).
R. Stanley, Enumerative combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999, p. 392, 7.24.3 Example.

Álvar Ibeas, First 25 rows, flattened
Wolfdieter Lang, Rows n=1..15 and color polynomials n=1..10.

Cf. A212357 for Z(C_n), A072605 for the row sums.

Cf. A000041 (row lengths), A036036, A185974, A212360, A213934, A318810.

a(n,k) is the number of necklace arrangements with n beads (respecting the cyclic C_n symmetry) with color assignment given by the multiset representative obtained uniquely from the k-th partition of n in A-St order. See the comment for more details and the A-St reference.
From Álvar Ibeas, Dec 12 2020: (Start)
Let L be the k-th partition of n in A-St and d be the gcd of its parts. Abusing the notation, we write a(n, L) for a(n, k) and accordingly for other partition arrays.
a(n, L) = n^(-1) * Sum_{v|d} phi(v) * A036038(n/v, L/v), where L/v is the partwise division of L by v.
a(n, L) = Sum_{v|d} A339677(L/v).
(End)
a(n,k) = A318810(A185974(n,k)). - Andrew Howroyd, Jan 23 2025

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Pontus von Brömssen, Sep 22 2024

nonn

a(n) is the number of occurrences of n in each of A036038, A050382, A078760, A318762, and A376367.

The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.

			a(6) = 3, because 6 can be written as a multinomial coefficient in 3 ways: 6 = 6!/(1!*5!) = 4!/(2!*2!) = 3!/(1!*1!*1!).

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A036038, A050382, A078760, A318762, A376367, A376368, A376370.

A096161 Row sums for triangle A096162.

Original entry on oeis.org

1, 3, 8, 30, 133, 768, 5221, 41302, 369170, 3677058, 40338310, 483134179, 6271796072, 87709287104, 1314511438945, 21017751750506, 357102350816602, 6424883282375340, 122025874117476166, 2439726373093186274

Offset: 1

Alford Arnold, Jun 18 2004

nonn

Also, partitions such that a set of k equal terms are labeled 1 through k and can appear in any order. For example, the partition 3+2+2+2+1+1+1+1 of 13 appears 1!*3!*4!=144 times because there are 1! ways to order the one "3," 3! ways to order the three "2"s, ... - Christian G. Bower, Jan 17 2006

			1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3 6 1 ... A036040
1 1 2 1 1 6 1 1 2 2 24 ... A096162
so a(n) begins 1 3 8 30 ... A096161

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

Cf. A005651, A000110, A036038, A036040, A096162, A110143, A287899.

nmax = 25; Rest[CoefficientList[Series[Product[Sum[k!*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2019 *)
m = 25; Rest[CoefficientList[Series[Product[-Gamma[0, -1/x^j] * Exp[-1/x^j], {j, 1, m}] / x^(m*(m + 1)/2), {x, 0, m}], x]] (* Vaclav Kotesovec, Dec 07 2020 *)

{ my(n=25); Vec(prod(k=1, n, O(x*x^n) + sum(r=0, n\k, x^(r*k)*r!))) }

G.f.: B(x)*B(x^2)*B(x^3)*... where B(x) is g.f. of A000142. - Christian G. Bower, Jan 17 2006
G.f.: Product_{k>0} Sum_{r>=0} x^(r*k)*r!. - Andrew Howroyd, Dec 22 2017
a(n) ~ n! * (1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + 587/n^7 + 3205/n^8 + 19201/n^9 + 123684/n^10 + ...), for coefficients see A293266. - Vaclav Kotesovec, Aug 10 2019

More terms from Vladeta Jovovic, Jun 22 2004

A096162 Let n be a number partitioned as n = b_1 + 2b_2 + ... + nb_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 1, 1, 2, 2, 24, 1, 1, 1, 2, 2, 6, 120, 1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720, 1, 1, 1, 1, 2, 1, 2, 2, 6, 2, 6, 24, 12, 120, 5040, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 6, 2, 4, 2, 24, 24, 6, 12, 120, 48, 720, 40320, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 6, 6, 2, 2, 2, 2, 6, 24, 6, 12, 4, 24, 120

Offset: 1

Alford Arnold, Jun 20 2004

The partitions of number n are grouped by increasing length and in reverse lexical order for partitions of the same length.

This sequence is in the Abramowitz-Stegun ordering, see A036036. - Hartmut F. W. Hoft, Apr 25 2015

			Illustrating the formula:
1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3  6  1 ... A036040
so
1 1 2 1 1 6 1 1 2  2 24 ... this sequence.
.
From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
The sequence as a structured triangle. The column headings indicate the number of elements in the underlying partitions. Brackets indicate groups of the products of factorials for all partitions of the same length when there is more than one partition.
     1   2        3        4     5    6
1:   1
2:   1   2
3:   1   1        6
4:   1  [1 2]     2       24
5:   1  [1 1]    [2 2]     6    120
6:   1  [1 1 2]  [2 1 6]  [6 4]  24  720
The partitions, their multiplicities and factorial products associated with the five entries in row n = 4 are:
partitions:         {4}, [{3, 1}, {2, 2}], {2, 1, 1}, {1, 1, 1, 1}
multiplicities:      1,  [{1, 1},  2],     {1, 2},     4
factorial products:  1!, [1!*1!, 2!],      1!*2!,      4!
(End)

Abramowitz and Stegun, Handbook of Mathematical Functions, p. 831, column "M_1" divided by "M_3."

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].

Row sums in A096161.
Row lengths in A000041.
Cf. A036038, A036040, A130561.

(* function a096162[ ] computes complete rows of the triangle *)
row[n_] := Map[Apply[Times, Map[Factorial, Last[Transpose[Tally[#]]]]]&, GatherBy[IntegerPartitions[n], Length], {2}]
triangle[n_] := Map[row, Range[n]]
a096162[n_] := Flatten[triangle[n]]
Take[a096162[9],90] (* data *)  (*Hartmut F. W. Hoft, Apr 25 2015 *)

from collections import Counter
def A096162_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [h(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (1..9): print(A096162_row(n)) # Peter Luschny, Nov 01 2019

T(n, k) = A036038(n,k) / A036040(n,k).

Appears to be n! / A130561(n); e.g., 4! / (24,24,12,12,1) = (1,1,2,2,24). - Tom Copeland, Nov 12 2017

Edited and extended by Christian G. Bower, Jan 17 2006

A183240 Sums of the squares of multinomial coefficients.

Original entry on oeis.org

1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, 180713279386, 18085215373130, 2188499311357525, 315204533416762046, 53270712928769375885, 10441561861586014363349, 2349364090881443819316871, 601444438364480313663234821, 173817677082622796179263021770

Offset: 0

Paul D. Hanna, Jan 03 2011

nonn

Equals sums of the squares of terms in rows of the triangle of multinomial coefficients (A036038).
Ignoring initial term, equals the logarithmic derivative of A183241; A183241 is conjectured to consist entirely of integers.
More generally, let {M(n,k), n>=0} be the sums of the k-th powers of the multinomial coefficients where k>=0 (see table A183610), then:
Sum_{n>=0} M(n,k)*x^n/n!^k = Product_{n>=1} 1/(1-x^n/n!^k).

			G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 46*x^3/3!^2 + 773*x^4/4!^2 +...
A(x) = 1/((1-x)*(1-x^2/2!^2)*(1-x^3/3!^2)*(1-x^4/4!^2)*...).
...
After the initial term a(0)=1, the next several terms are
a(1) = 1^2 = 1,
a(2) = 1^2 + 2^2 = 5,
a(3) = 1^2 + 3^2 + 6^2 = 46,
a(4) = 1^2 + 4^2 + 6^2 + 12^2 + 24^2 = 773,
a(5) = 1^2 + 5^2 + 10^2 + 20^2 + 30^2 + 60^2 + 120^2 = 19426,
and continue with the sums of squares of the terms in triangle A036038.

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A036038, A005651, A183235, A183236, A183237, A183238; A183241.

Cf. A183610 (table of sums of powers of multinomial coefficients).

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i))/i!^2+b(n, i-1))
    end:
a:= n-> n!^2*b(n$2):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 11 2019

t=Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i], Length[#1] > Length[#2] &]], {1}], {i, 30}]^2; Plus@@@t (* From Tony D. Noe *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
     b[n - i, Min[n - i, i]]/i!^2 + b[n, i - 1]];
a[n_] := n!^2 b[n, n];
a /@ Range[0, 21] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)

{a(n)=n!^2*polcoeff(1/prod(k=1,n,1-x^k/k!^2 +x*O(x^n)),n)}

G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = Product_{n>=1} 1/(1-x^n/n!^2).

a(n) ~ c * (n!)^2, where c = Product_{k>=2} 1/(1-1/(k!)^2) = 1.37391178018464563291052028168404977854977270679629932106310942272080844... . - Vaclav Kotesovec, Feb 19 2015

Terms following a(7) computed by T. D. Noe.

A376367 Sorted multinomial coefficients greater than 1, including duplicates.

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 30, 30, 31, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset: 1

Pontus von Brömssen, Sep 22 2024

nonn

Sorted terms of A036038, A050382, A078760, or A318762, excluding 1 (which appears infinitely often).

The number k appears A376369(k) times.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A036038, A050382, A078760, A318762, A376369, A376379.

a(n) = A318762(A376379(n)).

A049009 Number of functions from a set to itself such that the sizes of the preimages of the individual elements in the range form the n-th partition in Abramowitz and Stegun order.

Original entry on oeis.org

1, 1, 2, 2, 3, 18, 6, 4, 48, 36, 144, 24, 5, 100, 200, 600, 900, 1200, 120, 6, 180, 450, 300, 1800, 7200, 1800, 7200, 16200, 10800, 720, 7, 294, 882, 1470, 4410, 22050, 14700, 22050, 29400, 176400, 88200, 88200, 264600, 105840, 5040, 8, 448, 1568, 3136, 1960

Offset: 0

Alford Arnold

a(n,k) is a refinement of 1; 2,2; 3,18,6; 4,84,144,24; ... cf. A019575.
a(n,k)/A036040(n,k) and a(n,k)/A048996(n,k) are also integer sequences.
Apparently a(n,k)/A036040(n,k) = A178888(n,k). - R. J. Mathar, Apr 17 2011
Let f,g be functions from [n] into [n]. Let S_n be the symmetric group on n letters. Then f and g form the same partition iff S_nfS_n = S_ngS_n. - Geoffrey Critzer, Jan 13 2022

			Table begins:
  1;
  1;
  2,  2;
  3, 18,  6;
  4, 48, 36, 144, 24;
  ...
For n = 4, partition [3], we can map all three of {1,2,3} to any one of them, for 3 possible values. For n=5, partition [2,1], there are 3 choices for which element is alone in a preimage, 3 choices for which element to map that to and then 2 choices for which element to map the pair to, so a(5) = 3*3*2 = 18.

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page38.

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
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].

Cf. A019575, A035206, A035796, A036038, A036040, A048996.

Row sizes A000041, sums A000312.

f[list_] := Multinomial @@ Join[{nn - Length[list]}, Table[Count[list, i], {i, 1, nn}]]*Multinomial @@ list; Table[nn = n; Map[f, IntegerPartitions[nn]], {n, 0, 10}] // Grid (* Geoffrey Critzer, Jan 13 2022 *)

C(sig)={my(S=Set(sig)); (binomial(vecsum(sig), #sig)) * (#sig)! * vecsum(sig)! / (prod(k=1, #S, (#select(t->t==S[k], sig))!) * prod(k=1, #sig, sig[k]!))}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

a(n,k) = A036038(n,k) * A035206(n,k).

Better definition from Franklin T. Adams-Watters, May 30 2006

a(0)=1 prepended by Andrew Howroyd, Oct 18 2020

A157159 Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.

Original entry on oeis.org

1, 1, -1, 10, -16, 126, -526, 10312, -30024, 453840, -2805408, 45779328, -374664720, 7932770496, -67692115440, 2432120198016, -16610113920768, 437275706750208, -5110200130727808, 159305381515284480, -1931470594025607936, 63854116254680514048

Offset: 1

Wolfdieter Lang Mar 06 2009

sign

			Recurrence I: a(7) = 6! - (7*a(1)*a(6) + 21*a(2)*a(5) + 35*a(3)*a(4) + 105*a(1)*a(2)*a(4)) = 720 - (7*126 + 21*(-16) + 35*(-1)*10 + 105*10) = -526.
Recurrence II: a(4) = 3!*(1+2*(-1/2!)^2) + 1 = +10.

Alois P. Heinz, Table of n, a(n) for n = 1..170

Cf. A137852, A006973, A147542.

a:= proc(n) option remember; `if`(n=1, 1, (n-1)!*((-1)^n+add(d*
      (-a(d)/d!)^(n/d), d=numtheory[divisors](n) minus {1, n}))
       +(-1)^(n+1)*add((k-1)!*Stirling1(n, k), k=1..n))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 14 2012

a[n_] := a[n] = If[n == 1, 1, (n-1)!*((-1)^n+Sum[d*(-a[d]/d!)^(n/d), {d, Divisors[n][[2 ;; -2]]}])+(-1)^(n+1)*Sum[(k-1)!*StirlingS1[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Definition of a(n): 1-log(1-x) = product(1+a(n)*(x^n)/n!, n=1..infinity) (formal series and product).
Recurrence I. With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)) and the multinomial numbers M1(fp(n,m)) (given as array in A036038 for any partition) for fp(n,m) from FP(n,m): a(n) = (n-1)! - sum(sum(M1(fp)*product(a(k[j]),j=1..m),fp from FP(n,m)), m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=1, a(2)=1. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: a(n) = (n-1)!*((-1)^n + sum(d*(-a(d)/d!)^(n/d),d|n with 1A089064(n), n>=2, a(1)=1. A089064(n)=sum(((-1)^(m-1))*(m-1)!)*|S1(n,m)|, m=1..n) with the unsigned Stirling numbers of the first kind |A008275|. See the W. Lang link under A147542 for these recurrences.

More terms from Alois P. Heinz, Aug 14 2012

A325472 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

Original entry on oeis.org

1, 6, 10, 12, 15, 20, 21, 24, 28, 30, 35, 36, 42, 45, 55, 56, 60, 66, 70, 72, 78, 84, 90, 91, 105, 110, 120, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 210, 220, 231, 240, 252, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 360

Offset: 1

Alois P. Heinz, Sep 06 2019

nonn

Numbers that are repeated in the triangle A036038 (all positive integers occur at least once).

All triangular numbers (A000217) except 0 and 3 are in this sequence.

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
6 is in the sequence because M(6;5,1) = M(4;2,2) = M(3;1,1,1) = 6.
42 is in the sequence because M(42;41,1) = M(7;5,1,1) = 42.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A000041, A000217, A036038, A305188, A325306, A325593, A325901.

a(n) = A305188(n-1) for n > 1.

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