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A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m). 22
1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

FORMULA

Weigh transform of A000009.

EXAMPLE

4=(4)=(3)+(1)=(3+1)=(2+1)+(1).

From Gus Wiseman, Oct 11 2018: (Start)

a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:

  {{1}}  {{2}}  {{3}}      {{4}}        {{5}}        {{6}}

                {{1,2}}    {{1,3}}      {{1,4}}      {{1,5}}

                {{1},{2}}  {{1},{3}}    {{2,3}}      {{2,4}}

                           {{1},{1,2}}  {{1},{4}}    {{1,2,3}}

                                        {{2},{3}}    {{1},{5}}

                                        {{1},{1,3}}  {{2},{4}}

                                        {{2},{1,2}}  {{1},{1,4}}

                                                     {{1},{2,3}}

                                                     {{2},{1,3}}

                                                     {{3},{1,2}}

                                                     {{1},{2},{3}}

                                                     {{1},{2},{1,2}}

(End)

MAPLE

g:= proc(n, i) option remember; `if`(n=0, 1,

      `if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))

    end:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))

    end:

a:= n-> b(n, n):

seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

MATHEMATICA

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Dec 19 2015, after Alois P. Heinz *)

nn=10; Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k], {k, nn}], {x, 0, n}], {n, 0, nn}] (* Gus Wiseman, Oct 11 2018 *)

CROSSREFS

Cf. A050343-A050350, A089254.

Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.

Sequence in context: A158237 A117950 A025047 * A293642 A214286 A108700

Adjacent sequences:  A050339 A050340 A050341 * A050343 A050344 A050345

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1999

STATUS

approved

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Last modified November 22 18:55 EST 2019. Contains 329410 sequences. (Running on oeis4.)