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 *)
A115277
Number of partitions of {1,...,n} into blocks such that no even sized block is repeated.
Original entry on oeis.org
1, 1, 2, 5, 12, 37, 143, 562, 2320, 10941, 54865, 283890, 1604155, 9558226, 58668223, 384572975, 2631778832, 18576630237, 137919691717, 1060303298138, 8415786131309, 69538205444478, 591734670548037, 5194542789203877, 47127033586211659, 438972204436025198
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(
`if`(irem(i, 2)=0, 1, n), n/i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 08 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j! * b[n-i*j, i-1], {j, 0, Min[If[Mod[i, 2]==0, 1, n], n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)
A360182
Number of partitions of [n] where each block size occurs at most twice.
Original entry on oeis.org
1, 1, 2, 4, 14, 41, 152, 575, 2634, 13207, 59927, 312170, 1946870, 10547135, 65168469, 421552409, 3148178034, 20138277895, 141300123713, 1063603633154, 9108280640649, 68154636145922, 549824347467969, 4551458909818969, 39948625639349706, 406913301246314341
Offset: 0
a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 14: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 41: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 145|2|3, 14|25|3, 14|2|35, 15|24|3, 1|245|3, 1|24|35, 15|2|34, 1|25|34, 1|2|345.
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(combinat[multinomial](n, n-i*j, i$j)/j!*
b(n-i*j, i-1), j=0..min(2, n/i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, {n - i*j}~Join~ Table[i, {j}]]/j!*b[n - i*j, i - 1], {j, 0, Min[2, n/i]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}](* Jean-François Alcover, Nov 21 2023, after Alois P. Heinz *)
Showing 1-3 of 3 results.