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 *)
A229224
The partition function G(n,7).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4139, 21137, 115874, 677623, 4204927, 27565188, 190168577, 1376119903, 10414950785, 82230347149, 675762947626, 5768465148493, 51054457464731, 467728049807348, 4428770289719931, 43281554035140829, 436015324638219779
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, 7):
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, 7)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 22 2016
-
CoefficientList[Exp[Sum[x^j/j!, {j, 1, 7}]] + O[x]^25, x]*Range[0, 24]! (* Jean-François Alcover, May 21 2018 *)
A320763
Number of ordered set partitions of [n] where the maximal block size equals seven.
Original entry on oeis.org
1, 16, 288, 5280, 101640, 2068704, 44595408, 1017804216, 24559319070, 625388375040, 16772475939648, 472802783660064, 13981054231585584, 432866241024085440, 14006847775868101440, 472893544571144089536, 16631451859811919417144, 608402197372214335559040
Offset: 7
-
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))(7):
seq(a(n), n=7..25);
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);
Showing 1-4 of 4 results.
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