Search: a035649 -id:a035649
|
| |
|
|
A035648
|
|
Number of partitions of n into parts 6k+2 and 6k+5 with at least one part of each type.
|
|
+10
3
|
|
|
|
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 1, 3, 1, 4, 3, 7, 3, 8, 4, 10, 8, 14, 9, 17, 11, 22, 17, 28, 20, 34, 25, 43, 35, 53, 42, 64, 51, 80, 67, 96, 80, 115, 98, 142, 123, 168, 147, 200, 178, 244, 217, 286, 257, 339, 310, 407, 371, 475, 439, 559, 523, 664, 618, 772, 726
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,13
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 5))). - Robert Price, Aug 16 2020
|
|
|
MATHEMATICA
|
nmax = 68; s1 = Range[0, nmax/6]*6 + 2; s2 = Range[0, nmax/6]*6 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A035650
|
|
Number of partitions of n into parts 6k+3 and 6k+5 with at least one part of each type.
|
|
+10
3
|
|
|
|
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 3, 0, 1, 3, 1, 3, 6, 1, 3, 8, 3, 7, 12, 3, 9, 16, 7, 15, 22, 9, 19, 30, 16, 29, 40, 20, 38, 54, 32, 54, 69, 41, 70, 93, 61, 95, 118, 78, 124, 156, 110, 163, 195, 141, 211, 255, 192, 271, 317, 245, 349, 409, 323, 441, 506, 412, 562, 644
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,14
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 6))). - Robert Price, Aug 16 2020
|
|
|
MAPLE
|
b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
b(n, i-1, t, s)+(h-> `if`(h in {3, 5}, add(b(n-i*j, i-1,
`if`(h=3, 1, t), `if`(h=5, 1, s)), j=1..n/i), 0))(irem(i, 6))))
end:
a:= n-> b(n$2, 0$2):
|
|
|
MATHEMATICA
|
nmax = 71; s1 = Range[0, nmax/6]*6 + 3; s2 = Range[0, nmax/6]*6 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.006 seconds
|
