A175788
Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n that do not contain k as a part.
Original entry on oeis.org
1, 1, 1, 1, 0, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 2, 2, 2, 7, 1, 1, 2, 2, 3, 2, 11, 1, 1, 2, 3, 4, 4, 4, 15, 1, 1, 2, 3, 4, 5, 6, 4, 22, 1, 1, 2, 3, 5, 6, 8, 8, 7, 30, 1, 1, 2, 3, 5, 6, 9, 10, 11, 8, 42, 1, 1, 2, 3, 5, 7, 10, 12, 15, 15, 12, 56
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 1, 1, 1, 1, ...
2, 1, 1, 2, 2, 2, ...
3, 1, 2, 2, 3, 3, ...
5, 2, 3, 4, 4, 5, ...
7, 2, 4, 5, 6, 6, ...
Columns k=0-10 give:
A000041,
A002865,
A027336,
A027337,
A027338,
A027339,
A027340,
A027341,
A027342,
A027343,
A027344.
Main diagonal gives
A000065 (for n>0).
-
A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
A:= (n,k)-> A41(n) -`if`(k>0, A41(n-k), 0):
seq(seq(A(n,d-n), n=0..d), d=0..11);
-
A41[n_] := If[n<0, 0, PartitionsP[n]]; A[n_, k_] := A41[n]-If[k>0, A41[n-k], 0]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)
A343668
Number of partitions of an n-set without blocks of size 8.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4139, 21138, 115885, 677745, 4206172, 27577513, 190289713, 1377315050, 10426866782, 82350895629, 677003941219, 5781485704892, 51193839084907, 469251258854001, 4445769329586348, 43475305461354931, 438270620701587657, 4549243731200717053
Offset: 0
-
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(j=8, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Apr 25 2021
-
nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^8/8!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 8 k]/((n - 8 k)! k! (8!)^k), {k, 0, Floor[n/8]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n - 1, k - 1] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
A173304
Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 3, 4, 6, 4, 7, 8, 6, 1, 8, 11, 9, 2, 12, 15, 12, 3, 14, 20, 17, 5, 21, 26, 23, 7, 24, 35, 31, 11, 34, 45, 41, 15, 41, 58, 55, 21, 1, 55, 75, 71, 29, 1, 66, 96, 93, 40, 2, 88, 121, 120, 53, 3, 105, 154, 154, 72, 5, 137, 193, 196, 94, 7
Offset: 0
The finite difference array starts:
1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, ...; = A002865 (a variant)
1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, ...; = A027336
1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, ...; = A017338
1, 1, 2, 3, 5, 7, 11, ...; = A027342
...
Last, columns of the array become rows of triangle A173304:
1;
1;
1, 1;
2, 2, 1;
2, 3, 2;
4, 4, 3;
4, 6, 4, 1;
7, 8, 6, 1;
8, 11, 9, 2;
12, 15, 12, 3;
14, 20, 17, 5;
21, 26, 23, 7;
24, 35, 31, 11;
34, 45, 41, 15;
41, 58, 55, 21, 1;
55, 75, 71, 29, 1;
66, 96, 93, 40, 2;
88, 121, 120, 53, 3;
105, 154, 154, 72, 5;
137, 193, 196, 94, 7;
...
Showing 1-3 of 3 results.
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