A327729 -id:A327729 - cp's OEIS frontend

A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

1, 1, 1, 3, 6, 24, 84, 498, 2220, 15108, 92328, 773580, 5636460, 53563476, 471562512, 5270698716, 52117937052, 637276396764, 7317811499736, 100453675122444, 1276319138168796, 19048874583061716, 270233458572751440, 4442429353548965628, 68384217440167826412

Offset: 1

Alois P. Heinz, Sep 20 2019

nonn

Number of proper (n-1)-times partitions of n, cf. A327639.
Might be called "Half-Factorial numbers" analog to the "Half-Catalan numbers" (A000992).
The recursion formula is a special case of the formula given in A327729.
a(n+1)/(n*a(n)) tends to 0.67617164... - Vaclav Kotesovec, Apr 28 2020

			a(1) = 1:
  1
a(2) = 1:
  2 -> 11
a(3) = 1:
  3 -> 21 -> 111
a(4) = 3:
  4 -> 31 -> 211 -> 1111
  4 -> 22 -> 112 -> 1111
  4 -> 22 -> 211 -> 1111
a(5) = 6:
  5 -> 41 -> 311 -> 2111 -> 11111
  5 -> 41 -> 221 -> 1121 -> 11111
  5 -> 41 -> 221 -> 2111 -> 11111
  5 -> 32 -> 212 -> 1112 -> 11111
  5 -> 32 -> 212 -> 2111 -> 11111
  5 -> 32 -> 311 -> 2111 -> 11111

Alois P. Heinz, Table of n, a(n) for n = 1..481
Vaclav Kotesovec, Plot of a(n+1)/(n*a(n)) for n = 1..10000
Wikipedia, Partition (number theory)

Cf. A000142, A000992, A002846 (only one part of each size is replaceable), A327631, A327639, A327697, A327698, A327699, A327702, A327729.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> add(b(n$2, i)*(-1)^(n-1-i)*binomial(n-1, i), i=0..n-1):
seq(a(n), n=1..29);
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(j)*a(n-j)*binomial(n-2, j-1), j=1..n/2))
    end:
seq(a(n), n=1..29);

a[n_] := a[n] = Sum[Binomial[n-2, j-1] a[j] a[n-j], {j, n/2}]; a[1] = 1;
Array[a, 25] (* Jean-François Alcover, Apr 28 2020 *)

a(n) = Sum_{j=1..floor(n/2)} C(n-2,j-1) a(j)*a(n-j) for n > 1, a(1) = 1.

a(n) = A327639(n,n-1) = A327631(n,n-1)/n.

A327711 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n (k is a partition length).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 27, 55, 171, 475, 1555, 4915, 20023, 68243, 288024, 1213828, 5435935, 23966970, 121432923, 578757824, 3130381590, 16427772974, 91877826663, 519546134163, 3199523135912, 18868494152257, 120274458082095, 772954621249540, 5219747666882153

Offset: 0

Alois P. Heinz, Sep 22 2019

nonn

Number of partitions of [n] whose block sizes are nondecreasing when blocks are ordered by their minima and these minima are {1..k} (for some k <= n). a(5) = 10: 12345, 13|245, 14|235, 15|234, 1|2345, 1|24|35, 1|25|34, 1|2|345, 1|2|3|45, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..635
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A005651, A179973, A326493, A327712, A327729.

with(combinat):
a:= n-> add(multinomial(n-nops(p), map(
    x-> x-1, p)[], 0), p=partition(n)):
seq(a(n), n=0..28);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<2, 0,
      b(n, i-1, p)) +b(n-i, min(n-i, i), p-1)/(i-1)!)
    end:
a:= n-> b(n$3):
seq(a(n), n=0..28);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 2, 0, b[n, i - 1, p]] + b[n - i, Min[n - i, i], p - 1]/(i - 1)!];
a[n_] := b[n, n, n];
a /@ Range[0, 28] (* Jean-François Alcover, May 01 2020, from 2nd Maple program *)

cp's OEIS Frontend

A327643 Number of refinement sequences n -> ... -> {1}^n, where in each step one part is replaced by a partition of itself into two smaller parts (in weakly decreasing order).

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