A115276
Number of partitions of {1,...,n} into block sizes not a multiple of 4.
Original entry on oeis.org
1, 1, 2, 5, 14, 47, 173, 702, 3124, 14901, 76405, 417210, 2411466, 14731095, 94573911, 636575050, 4480990936, 32887804361, 251236573561, 1993395483746, 16397468177406, 139634290253907, 1229013163330947, 11166172488138322, 104593176077399652
Offset: 0
-
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
irem(j, 4)=0, 0, binomial(n-1, j-1)*a(n-j)), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 17 2015
-
a[n_] := a[n] = If[n == 0, 1, Sum[If[Mod[j, 4] == 0, 0, Binomial[n - 1, j - 1]*a[n - j]], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
A115278
Number of partitions of {1,...,2*n} into even sized blocks such that no block size is repeated.
Original entry on oeis.org
1, 1, 1, 16, 29, 256, 14422, 49141, 490429, 10758400, 1797335306, 9458619391, 133756636598, 2528529510391, 137864810180749, 53441183229799381, 410251032050409469, 7615997734377068128, 167055180095977694194, 6741819165851219788075, 738863335901972011745434
Offset: 0
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
end:
a:= n-> b(2*n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 08 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!* b[n - i*j, i - 2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[2 n, 2 n]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)
A115275
Number of partitions of {1,...,n} into blocks such that no block size is repeated more than 3 times.
Original entry on oeis.org
1, 1, 2, 5, 14, 51, 187, 820, 3670, 18191, 97917, 554500, 3334465, 20871592, 138440031, 972083845, 6985171390, 52194795327, 412903730293, 3313067916192, 28017395030419, 241504438776956, 2189375704925081, 19771679215526507, 187677937412341677
Offset: 0
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1), j=0..min(3, n/i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Sep 17 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-1], {j, 0, Min[3, n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)
Showing 1-3 of 3 results.