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A320553 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most three elements and for at least one block c the smallest integer interval containing c has exactly three elements. 3
2, 5, 12, 29, 66, 145, 315, 676, 1436, 3031, 6367, 13323, 27798, 57873, 120281, 249657, 517663, 1072520, 2220724, 4595938, 9508022, 19664296, 40659943, 84057614, 173750589, 359110196, 742150185, 1533651213, 3169118648, 6548358736, 13530454573, 27956404705 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

LINKS

Alois P. Heinz, Table of n, a(n) for n = 3..1000

FORMULA

G.f.: (x+2)*x^3/((x^2+x-1)*(x^4+2*x^3+x^2+x-1)).

a(n) = A129847(n) - A000045(n+1).

EXAMPLE

a(4) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.

MAPLE

b:= proc(n, m, l) option remember; `if`(n=0, 1,

      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],

      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))

    end:

A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):

a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(3):

seq(a(n), n=3..35);

MATHEMATICA

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1]~Complement~{0}}]];

A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];

a[n_] := With[{k = 3}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];

a /@ Range[3, 35] (* Jean-Fran├žois Alcover, Dec 08 2020, after Alois P. Heinz *)

CROSSREFS

Column k=3 of A276727.

Cf. A000045, A129847, A320616.

Sequence in context: A232534 A274594 A062422 * A266465 A079864 A131045

Adjacent sequences:  A320550 A320551 A320552 * A320554 A320555 A320556

KEYWORD

nonn,easy

AUTHOR

Alois P. Heinz, Oct 15 2018

STATUS

approved

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Last modified July 7 12:47 EDT 2022. Contains 355148 sequences. (Running on oeis4.)