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A248518 Number of partitions of n into parts > 0 without 1 as digit, cf. A052383. 2
1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 34, 44, 51, 66, 75, 96, 110, 136, 157, 193, 220, 267, 307, 367, 421, 501, 571, 677, 772, 905, 1033, 1207, 1371, 1595, 1812, 2096, 2377, 2741, 3101, 3564, 4028, 4608, 5203, 5938, 6688, 7612, 8564, 9719, 10919 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Note that the definition says "1 as a DIGIT", not "1 as a PART". - N. J. A. Sloane, Jun 28 2017
LINKS
EXAMPLE
The full list of partitions of 10 is as follows:
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 2],
[1, 1, 1, 1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2, 2], [1, 1, 2, 2, 2, 2],
[2, 2, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 2, 3],
[1, 1, 1, 2, 2, 3], [1, 2, 2, 2, 3], [1, 1, 1, 1, 3, 3], [1, 1, 2, 3, 3],
[2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 2, 4],
[1, 1, 2, 2, 4], [2, 2, 2, 4], [1, 1, 1, 3, 4], [1, 2, 3, 4], [3, 3, 4],
[1, 1, 4, 4], [2, 4, 4], [1, 1, 1, 1, 1, 5], [1, 1, 1, 2, 5], [1, 2, 2, 5],
[1, 1, 3, 5], [2, 3, 5], [1, 4, 5], [5, 5], [1, 1, 1, 1, 6], [1, 1, 2, 6],
[2, 2, 6], [1, 3, 6], [4, 6], [1, 1, 1, 7], [1, 2, 7], [3, 7], [1, 1, 8],
[2, 8], [1, 9], [10]]
If we excluse those that have a 1 in one of the parts, 11 partitions are left:
[[2, 2, 2, 2, 2], [2, 2, 3, 3], [2, 2, 2, 4], [3, 3, 4], [2, 4, 4], [2, 3, 5], [5, 5], [2, 2, 6], [4, 6], [3, 7], [2, 8]].
So a(10) = 11. - N. J. A. Sloane, Jun 28 2017
a(11) = #[9+2, 8+3, 7+4, 7+2+2, 6+5, 6+3+2, 5+4+2, 5+3+3, 5+2+2+2, 4+4+3, 4+3+2+2, 3+3+3+2, 3+2+2+2+2} = 13;
a(12) = #{9+3, 8+4, 8+2+2, 7+5, 7+3+2, 6+6, 6+4+2, 6+3+3, 6+2+2+2, 5+5+2, 5+4+3, 5+3+2+2, 4+4+4, 4+4+2+2, 4+3+3+2, 4+2+2+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 19.
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[Flatten[ IntegerDigits/@#], 1]&]], {n, 0, 60}] (* Harvey P. Dale, Jun 28 2017 *)
PROG
(Haskell)
a248518 = p $ tail a052383_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
CROSSREFS
Sequence in context: A060029 A100471 A266777 * A095700 A339404 A035944
KEYWORD
nonn,base
AUTHOR
Reinhard Zumkeller, Oct 07 2014
STATUS
approved

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