A005651 -id:A005651 - cp's OEIS frontend

A183239 G.f.: exp( Sum_{n>=1} A005651(n)*x^n/n ), where A005651 gives the sums of multinomial coefficients.

1, 1, 2, 5, 17, 69, 352, 2077, 14505, 114354, 1023839, 10130051, 110878314, 1320375213, 17086334702, 237832320231, 3552995476517, 56590659564489, 958653346775294, 17192978984630744, 325681548343314833, 6494280460641306608

Offset: 0

Paul D. Hanna, Jan 03 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 69*x^5 + 352*x^6 +...
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 47*x^4/4 + 246*x^5/5 + 1602*x^6/6 + 11481*x^7/7 + 95503*x^8/8 +...+ A005651(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A005651, A183238, A183241.

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

a(n) ~ c * (n-1)!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264... . - Vaclav Kotesovec, Feb 19 2015

A097522 Triangle read by rows giving the 246 multinomials described by A005651(5) related to Young tableau and Kostka numbers.

Original entry on oeis.org

1, 16, 1, 25, 12, 1, 36, 15, 8, 1, 25, 18, 10, 8, 1, 16, 10, 6, 5, 4, 1, 1, 4, 5, 6, 5, 4, 1

Offset: 1

Alford Arnold, Aug 27 2004

The antidiagonal 1 4 5 6 5 4 1 is also listed in A007837 in ascending order by value: 1 1 4 4 5 5 6. The 246 cases are distributed in A036038 as 1 5 10 20 30 60 120.

			Triangle is
   1;
  16,  1;
  25, 12,  1;
  36, 15,  8,  1;
  25, 18, 10,  8,  1;
  16, 10,  6,  5,  4,  1;
   1,  4,  5,  6,  5,  4,  1;

R. Stanton, Constructive Combinatorics, 19856, page 83.

Cf. A000085, A000142, A003869, A005651, A036038, A060240.

A104707 Triangle read by rows distributing the 1602 multinomials described by A005651(6) related to Young tableau and Kostka numbers.

Original entry on oeis.org

1, 25, 1, 81, 20, 1, 25, 54, 15, 1, 100, 15, 36, 15, 1, 256, 60, 10, 27, 10, 1, 25, 128, 30, 5, 27, 10, 1, 100, 10, 64, 30, 5, 18, 10, 1, 81, 40, 5, 32, 10, 5, 9, 5, 1, 25, 27, 10, 0, 32, 10, 0, 9, 5, 1, 1, 5, 9, 10, 5, 16, 10, 5, 9, 5, 1

Offset: 1

Alford Arnold, Mar 19 2005

The last row (and the square roots of the first column) is 1 5 9 5 10 16 5 10 9 5 1, which when sorted appears as 1 1 5 5 5 5 9 9 10 10 16 in A003870 and A060240. The 1602 cases are distributed in A036038: 1 6 15 20 30 60 90 120 180 360 720.

			The triangle is:
1;
25,   1;
81,  20,  1;
25,  54, 15,  1;
100, 15, 36, 15,  1;
256, 60, 10, 27, 10,  1;
25, 128, 30,  5, 27, 10,  1;
100, 10, 64, 30,  5, 18, 10, 1;
81,  40,  5, 32, 10,  5,  9, 5, 1;
25,  27, 10,  0, 32, 10,  0, 9, 5, 1;
1,    5,  9, 10,  5, 16, 10, 5, 9, 5, 1;

D. Stanton and D. White, Constructive Combinatorics, 1986, page 83.

Cf. A000041, A000085, A000142, A003870, A060240, A005651, A036038 and A097522 (a similar triangle distributing the 246 multinomials in A005651(5)).

Cf. A104778.

A088946 Coefficient of x^n of A(x)^n is A005651(n), which is the sum of multinomial coefficients for n.

Original entry on oeis.org

1, 1, 1, 1, 4, 13, 86, 466, 3830, 29633, 289480, 2865826, 32992576, 394893590, 5267005771, 73816682108, 1123682376634, 18021084739737, 309127486471294, 5578794711931042, 106617559112195219

Offset: 0

Paul D. Hanna, Oct 25 2003

nonn

Cf. A005651.

A007837 Number of partitions of n-set with distinct block sizes.

Original entry on oeis.org

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847

Offset: 0

Arnold Knopfmacher

nonn

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..700
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Fig. 3, arXiv:math/0606370 [math.CO], 2006.
Knopfmacher, A., Odlyzko, A. M., Pittel, B., Richmond, L. B., Stark, D., Szekeres, G. and Wormald, N. C., The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin., 6 (1999), no. 1, Research Paper 2, 36 pp.
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011

nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)

{my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

E.g.f.: Product_{m >= 1} (1+x^m/m!).
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) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Christian G. Bower

a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A036038 Triangle of multinomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720, 1, 7, 21, 35, 42, 105, 140, 210, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, 56, 168, 280, 420, 560, 336, 840, 1120, 1680, 2520, 1680, 3360, 5040, 6720

Offset: 1

N. J. A. Sloane

The number of terms in the n-th row is the number of partitions of n, A000041(n). - Amarnath Murthy, Sep 21 2002
For each n, the partitions are ordered according to A-St: first by length and then lexicographically (arranging the parts in nondecreasing order), which is different from the usual practice of ordering all partitions lexicographically. - T. D. Noe, Nov 03 2006
For this ordering of the partitions, for n >= 1, see the remarks and the C. F. Hindenburg link given in A036036. - Wolfdieter Lang, Jun 15 2012
The relation (n+1) * A134264(n+1) = A248120(n+1) / a(n) where the arithmetic is performed for matching partitions in each row n connects the combinatorial interpretations of this array to some topological and algebraic constructs of the two other entries. Also, these seem (cf. MOPS reference, Table 2) to be the coefficients of the Jack polynomial J(x;k,alpha=0). - Tom Copeland, Nov 24 2014
The conjecture on the Jack polynomials of zero order is true as evident from equation a) on p. 80 of the Stanley reference, suggested to me by Steve Kass. The conventions for denoting the more general Jack polynomials J(n,alpha) vary. Using Stanley's convention, these Jack polynomials are the umbral extensions of the multinomial expansion of (s_1*x_1 + s_2*x_2 + ... + s_(n+1)*x_(n+1))^n in which the subscripts of the (s_k)^j in the symmetric monomial expansions are finally ignored and the exponent dropped to give s_j(alpha) = j-th row polynomial of A094638 or |A008276| in ascending powers of alpha. (The MOPS table has some inconsistency between n = 3 and n = 4.) - Tom Copeland, Nov 26 2016

			1;
1, 2;
1, 3,  6;
1, 4,  6, 12, 24;
1, 5, 10, 20, 30, 60, 120;
1, 6, 15, 20, 30, 60,  90, 120, 180, 360, 720;

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".

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.
T. Copeland, Witt Differential Generator for Special Jack Symmetric Functions / Polynomials, 2016.
I. Dumitriu, A. Edelman, G. Schuman, MOPS: Multivariate orthogonal polynomials (symbolically), arxiv:0409066 [math-ph], 2004.
Wolfdieter Lang, First 10 rows and more.
R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. in Math., 77, p. 76-115, 1989.

Cf. A036036-A036040. Different from A078760. Row sums give A005651.
Cf. A183610 is a table of sums of powers of terms in rows.
Cf. A134264 and A248120.
Cf. A008275, A008276, A094638, A248927.
Cf. A096162 for connections to A130561.

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 sA036038(n, m) := n!/ (mul((t!)^q(t), t=1..n)); od: od: seq(seq(A036038(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

Flatten[Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i],  Length[ #1]>Length[ #2]&]], {1}], {i,9}]] (* T. D. Noe, Nov 03 2006 *)

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 A036038_row(n):
    return [multinomial(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A036038_row(n))
# Peter Luschny, Dec 18 2016, corrected Apr 30 2022

The n-th row is the expansion of (x_1 + x_2 + ... + x_(n+1))^n in the basis of the monomial symmetric polynomials (m.s.p.). E.g., (x_1 + x_2 + x_3 + x_4)^3 = m[3](x_1,..,x_4) + 3*m[1,2](x_1,..,x_4) + 6*m[1,1,1](x_1,..,x_4) = (Sum_{i=1..4} x_i^3) + 3*(Sum_{i,j=1..4;i != j} x_i^2 x_j) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k). The number of indeterminates can be increased indefinitely, extending each m.s.p., yet the expansion coefficients remain the same. In each m.s.p., unique combinations of exponents and subscripts appear only once with a coefficient of unity. Umbral reduction by replacing x_k^j with x_j in the expansions gives the partition polynomials of A248120. - Tom Copeland, Nov 25 2016
From Tom Copeland, Nov 26 2016: (Start)
As an example of the umbral connection to the Jack polynomials: J(3,alpha) = (Sum_{i=1..4} x_i^3)*s_3(alpha) + 3*(Sum_{i,j=1..4;i!=j} x_i^2 x_j)*s_2(alpha)*s_1(alpha)+ 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k)*s_1(alpha)*s_1(alpha)*s_1(alpha) = (Sum_{i=1..4} x_i^3)*(1+alpha)*(1+2*alpha)+ 3*(sum_{i,j=1..4;i!=j} x_i^2 x_j)*(1+alpha) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k).
See the Copeland link for more relations between the multinomial coefficients and the Jack symmetric functions. (End)

More terms from David W. Wilson and Wouter Meeussen

A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 1, 5, 10, 1, 1, 15, 1, 10, 15, 6, 1, 10, 10, 7, 15, 15, 1, 60, 1, 1, 21, 8, 35, 45, 1, 9, 28, 20, 1, 105, 1, 21, 105, 10, 1, 15, 35, 70, 36, 28, 1, 105, 56, 35, 45, 11, 1, 210, 1, 12, 210, 1, 84, 168, 1, 36, 55, 280, 1, 105, 1, 13, 280, 45, 126, 252, 1

Offset: 1

Max Alekseyev, Nov 07 2006

nonn

Coefficients of (D^k f)(g(t))*(D g(t))^k1*(D^2 g(t))^k2*... in the Faa di Bruno formula for D^m(f(g(t))) where k = k1 + k2 + ..., m = 1*k1 + 2*k2 + ....

Number of set partitions whose block sizes are the prime indices of n (i.e., the integer partition with Heinz number n). - Gus Wiseman, Sep 12 2018

			The a(6) = 3 set partitions of type (2,1) are {{1},{2,3}}, {{1,3},{2}}, {{1,2},{3}}. - _Gus Wiseman_, Sep 12 2018

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Faà di Bruno's Formula

Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.

Cf. A065475, A100484.

with(numtheory):
a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i],
         i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)),
         i=1..pi(max(1, factorset(n))))]):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 14 2020

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[numSetPtnsOfType[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}] (* Gus Wiseman, Sep 12 2018 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k,1])*f[k,2])!/(prod(k=1, #f~, f[k,2]!)*prod(k=1, #f~, primepi(f[k,1])!^f[k,2])); \\ Michel Marcus, Oct 11 2023

For n = p1^k1*p2^k2*... where 2 = p1 < p2 < ... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1!^k1*2!^k2*...)).

a(2*prime(n)) = n + 1, for n > 1. See A065475. - Bill McEachen, Oct 11 2023

A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 1, 3, 6, 6, 4, 5, 1, 1, 2, 2, 2, 6, 3, 3, 3, 4, 4, 12, 10, 5, 6, 1, 1, 2, 2, 1, 3, 2, 3, 6, 6, 3, 1, 12, 4, 12, 6, 10, 5, 20, 15, 6, 7, 1, 1, 2, 2, 2, 3, 2, 6, 3, 3, 4, 6, 6, 1, 12, 12, 4, 12

Offset: 0

Gus Wiseman, Sep 13 2018

A refinement of Pascal's triangle, these are the unsigned coefficients appearing in the expansion of homogeneous symmetric functions in terms of elementary symmetric functions.

			Triangle begins:
  1
  1
  1  1
  1  2  1
  1  1  2  3  1
  1  2  2  3  3  4  1
  1  2  2  1  1  3  6  6  4  5  1
The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).

Alois P. Heinz, Rows n = 0..33, flattened

A different row ordering is A072811.

Cf. A000041, A005651, A008277, A008480, A056239, A124794, A215366, A319182, A319191, A319192.

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(
      map(p-> p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> map(m-> (l-> add(i, i=l)!/mul(i!, i=l))(map(
        i-> i[2], ifactors(m)[2])), sort(b(n$2)))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Permutations[primeMS[k]]],{n,6},{k,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0 || i < 2, {2^n}, Flatten[Table[ #*Prime[i]^j& /@ b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Map[Function[m, Function[l, Total[l]!/Times @@ (l!)][ FactorInteger[m][[All, 2]]]], Sort[b[n, n]]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

T(n,k) = A008480(A215366(n,k)).

T(0,1)=1 prepended by Alois P. Heinz, Feb 14 2020

A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 1, 2, -3, -6, 1, 3, 8, 24, -6, -120, -30, -20, 1, 720, 15, -5040, 20, 90, 144, 40320, -10, 40, -840, -15, -90, -362880, -120, 3628800, 1, -504, 5760, -420, 45, -39916800, -45360, 3360, 40, 479001600, 630, -6227020800, 504, 210, 403200, 87178291200

Offset: 1

Gus Wiseman, Sep 13 2018

sign

A refinement of Stirling numbers of the first kind.

An unsigned version is A124795.

Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]],{n,100}]

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (-1)^(Sum x_i * y_i - Sum y_i) (Sum x_i * y_i)! / (Product x_i^y_i * Product y_i!).

A141199 Number of hierarchical ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 7, 17, 38, 87, 191, 421, 911, 1963, 4186, 8885, 18724, 39284, 82005, 170521, 353214, 729290, 1501184, 3081869, 6311404, 12896983, 26301515, 53541702, 108815626, 220824295, 447524559, 905850001, 1831526719

Offset: 0

Thomas Wieder, Jun 13 2008, Jun 29 2008, Jul 28 2008

nonn

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).
There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.
Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.
With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);
The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.
In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - Gus Wiseman, Dec 05 2022

			n=1:
[1]
-------------------------
n=2:
[1, 1],
[1, "|", 1],
[2]
-------------------------
n=3:
[1, 2],
[1, "|", 1, "|", 1],
[1, 1, 1],
[3],
[2, "|", 1],
[1, 1, "|", 1],
[1, "|", 2]
-------------------------
n=4:
[1, 1, 1, "|", 1],
[1, 1, "|", 1, 1],
[2, 2],
[1, 3],
[1, 1, 1, 1],
[1, 1, 2],
[4],
[1, "|", 1, "|", 1, "|", 1],
[1, 2, "|", 1],
[1, 1, "|", 2],
[1, 1, "|", 1, "|", 1],
[2, "|", 1, "|", 1],
[1, "|", 2, "|", 1],
[1, "|", 1, "|", 2],
[1, "|", 3],
[3, "|", 1],
[2, "|", 2].

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [_Thomas Wieder_, Nov 14 2009]

Cf. A055887, A083355, A140585.

Cf. A000041, A000219, A001970, A005651, A063834, A129519, A218482, A358335, A358831, A358836, A358908.

A Maple program to generate these "hierarchically ordered partitions of partitions" is available on request.
An asymptotic expansion can be found using the generating function given by Vladeta Jovovic. For that purpose we use the Maple program "equivalent" from Bruno Salvy (http://ago.inria.fr/libraries/libraries.html).

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 18 2022

G.f.: 1/Product_{i>=1} (1-x^i/Product_{j=1..i} (1-x^j)). - Vladeta Jovovic, Jul 16 2008

More terms from Vladeta Jovovic, Jul 16 2008

a(0)=1 prepended by Seiichi Manyama, Jan 18 2022

cp's OEIS Frontend

A183239 G.f.: exp( Sum_{n>=1} A005651(n)*x^n/n ), where A005651 gives the sums of multinomial coefficients.

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A097522 Triangle read by rows giving the 246 multinomials described by A005651(5) related to Young tableau and Kostka numbers.

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A104707 Triangle read by rows distributing the 1602 multinomials described by A005651(6) related to Young tableau and Kostka numbers.

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A088946 Coefficient of x^n of A(x)^n is A005651(n), which is the sum of multinomial coefficients for n.

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A007837 Number of partitions of n-set with distinct block sizes.

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A036038 Triangle of multinomial coefficients.

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A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

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A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).

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