0,4

This is a function of the individual partitions of an integer. The number of values in each line is given by A000041; thus lines 0 to 5 of the sequence are (1), (1), (1,2), (1,3,6), (1,4,6,12,24). The partitions in each line are ordered with the largest part sizes first, so the line 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1]. Note that exponents are often used to represent repeated values in a partition, so the last index could instead be written [1^4]. The combination function (sequence A007318) C(n,m) = C([m,n-m]).

This sequence is also the sequence of multinomial coefficients for partitions ordered lexicographically, matching partition sequence A080577. This is different ordering than in sequence A036038 of multinomial coefficients. - Sergei Viznyuk, Mar 15 2012

T. D. Noe, Rows n=0..25 of triangle, flattened

S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.

Sergei Viznyuk, C Program

Index entries for triangles and arrays related to Pascal's triangle.

C([<a(i)>]) = (Sum a(i))! / Product a(i) !.

T(n,k) = A008480(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

The irregular table starts:

[0] {1},

[1] {1},

[2] {1, 2},

[3] {1, 3, 6},

[4] {1, 4, 6, 12, 24},

[5] {1, 5, 10, 20, 30, 60, 120},

[6] {1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720}

...

C([2,1]) = 3 for the labelings ({1,2},{3}), ({1,3},{2}) and ({2,3},{2}).

g:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):

b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->

[i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:

seq(T(n), n=0..9); # Alois P. Heinz, Mar 25 2020

Flatten[Table[Apply[Multinomial, IntegerPartitions[i], {1}], {i, 0, 25}]] (* T. D. Noe, Oct 14 2007 *)

Flatten[ Multinomial @@@ IntegerPartitions @ # & /@ Range[ 0, 8]] (* Michael Somos, Feb 05 2011 *)

g[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times@@(ee!)];

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i] & /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];

row[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}] & /@ b[n, n];

T[n_] := g /@ row[n];

T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

(PARI)

C(sig)={vecsum(sig)!/vecprod(apply(k->k!, sig))}

Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}

{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

(SageMath)

def A070289_row(n): return [multinomial(x) for x in Partitions(n)]

print(flatten([A070289_row(n) for n in range(8)])) # Peter Luschny, Jun 24 2025

Different from A036038.

Cf. A000041, A008480, A063008, A080577.

Sequence in context: A171999 A036038 A210237 * A348113 A103280 A046899

Adjacent sequences: A078757 A078758 A078759 * A078761 A078762 A078763

nice,easy,nonn,tabf,look

Franklin T. Adams-Watters, Jan 08 2003

approved