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).
Original entry on oeis.org
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
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
-
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 *)
A327697
Number of refinement sequences n -> ... -> {1}^n, where in each step every single part of a nonempty selection of parts is replaced by a partition of itself into smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 2, 7, 22, 122, 598, 4683, 31148, 292008, 2560274, 30122014, 313694962, 4189079688, 53048837390, 826150653479, 11827659365138, 204993767192252, 3371451881544534, 65337695492942258, 1198123466804343518, 25318312971995895392, 516420623159289735874
Offset: 1
a(1) = 1:
1
a(2) = 1:
2 -> 11
a(3) = 2:
3 -> 111
3 -> 21 -> 111
a(4) = 7:
4 -> 1111
4 -> 211 -> 1111
4 -> 31 -> 1111
4 -> 31 -> 211 -> 1111
4 -> 22 -> 1111
4 -> 22 -> 112 -> 1111
4 -> 22 -> 211 -> 1111
A327698
Number of refinement sequences n -> ... -> {1}^n, where in each step exactly one part is replaced by a partition of itself into smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 2, 6, 17, 74, 300, 1755, 9360, 65510, 442117, 3802889, 30213386, 294892947, 2789021105, 31360525517, 334374848070, 4184958056248, 50606351991305, 704124800141153, 9452367941048830, 143309007303310536, 2124982437997726705, 35389562541842450218
Offset: 1
a(4) = 6:
4 -> 1111
4 -> 211 -> 1111
4 -> 31 -> 1111
4 -> 31 -> 211 -> 1111
4 -> 22 -> 112 -> 1111
4 -> 22 -> 211 -> 1111
A327699
Number of refinement sequences n -> ... -> {1}^n, where in each step every single part of a nonempty selection of parts is replaced by a partition of itself into two smaller parts (in weakly decreasing order).
Original entry on oeis.org
1, 1, 1, 4, 9, 48, 211, 1736, 9777, 91169, 739174, 8613817, 83763730, 1105436491, 13222076337, 207852246589, 2789691577561, 47759515531854, 755158220565169, 14595210284816038, 255814560447492788, 5373613110108953192, 105867623217924984398, 2460702471446564481641
Offset: 1
a(4) = 4:
4 -> 31 -> 211 -> 1111
4 -> 22 -> 1111
4 -> 22 -> 112 -> 1111
4 -> 22 -> 211 -> 1111
Showing 1-4 of 4 results.
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