A276921
Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 6, 0, 1, 1, 3, 12, 24, 0, 1, 1, 3, 13, 66, 120, 0, 1, 1, 3, 13, 74, 450, 720, 0, 1, 1, 3, 13, 75, 530, 3690, 5040, 0, 1, 1, 3, 13, 75, 540, 4550, 35280, 40320, 0, 1, 1, 3, 13, 75, 541, 4670, 45570, 385560, 362880, 0
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 2, 3, 3, 3, 3, 3, 3, ...
0, 6, 12, 13, 13, 13, 13, 13, ...
0, 24, 66, 74, 75, 75, 75, 75, ...
0, 120, 450, 530, 540, 541, 541, 541, ...
0, 720, 3690, 4550, 4670, 4682, 4683, 4683, ...
0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...
Columns k=0..10 give:
A000007,
A000142,
A080599,
A189886,
A276924,
A276925,
A276926,
A276927,
A276928,
A276929,
A276930.
-
A:= proc(n, k) option remember; `if`(n=0, 1, add(
A(n-i, k)*binomial(n, i), i=1..min(n, k)))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)
A229225
The partition function G(n,8).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115964, 678448, 4212352, 27632112, 190778186, 1381763398, 10468226150, 82744297014, 680835331228, 5819712427654, 51584619782546, 473344099095848, 4489677962922186, 43957668431564086, 443694809361207824
Offset: 0
-
G:= proc(n, k) option remember; local j; if k>n then G(n, n)
elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
end:
a:= n-> G(n, 8):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-i)*binomial(n-1, i-1), i=1..min(n, 8)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 22 2016
-
CoefficientList[Exp[Sum[x^j/j!, {j, 1, 8}]] + O[x]^25, x]*Range[0, 24]! (* Jean-François Alcover, May 21 2018 *)
A320764
Number of ordered set partitions of [n] where the maximal block size equals eight.
Original entry on oeis.org
1, 18, 360, 7260, 152460, 3361644, 78041964, 1908389340, 49118959890, 1328964080730, 37738620245898, 1122927974067042, 34953464391146730, 1136306352798186570, 38520124906043253330, 1359621561034260858906, 49896547074800880656202, 1901350452285623246965200
Offset: 8
-
b:= proc(n, k) option remember; `if`(n=0, 1, add(
b(n-i, k)*binomial(n, i), i=1..min(n, k)))
end:
a:= n-> (k-> b(n, k) -b(n, k-1))(8):
seq(a(n), n=8..25);
A320765
Number of ordered set partitions of [n] where the maximal block size equals nine.
Original entry on oeis.org
1, 20, 440, 9680, 220220, 5229224, 130069940, 3392692160, 92780281880, 2657929522820, 79670485645608, 2495398380120360, 81558207395885220, 2777643033619233780, 98440545801322467600, 3625667341827832048176, 138601954935720474004950, 5492809832014657114548300
Offset: 9
-
b:= proc(n, k) option remember; `if`(n=0, 1, add(
b(n-i, k)*binomial(n, i), i=1..min(n, k)))
end:
a:= n-> (k-> b(n, k) -b(n, k-1))(9):
seq(a(n), n=9..25);
Showing 1-4 of 4 results.
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