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A248518 Number of partitions of n into parts > 0 without 1 as digit, cf. A052383.

%I A248518 #17 Nov 11 2024 20:31:20
%S A248518 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,
%T A248518 220,267,307,367,421,501,571,677,772,905,1033,1207,1371,1595,1812,
%U A248518 2096,2377,2741,3101,3564,4028,4608,5203,5938,6688,7612,8564,9719,10919
%N A248518 Number of partitions of n into parts > 0 without 1 as digit, cf. A052383.
%C A248518 Note that the definition says "1 as a DIGIT", not "1 as a PART". - _N. J. A. Sloane_, Jun 28 2017
%e A248518 The full list of partitions of 10 is as follows:
%e A248518 [[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 2],
%e A248518     [1, 1, 1, 1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2, 2], [1, 1, 2, 2, 2, 2],
%e A248518     [2, 2, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 2, 3],
%e A248518     [1, 1, 1, 2, 2, 3], [1, 2, 2, 2, 3], [1, 1, 1, 1, 3, 3], [1, 1, 2, 3, 3],
%e A248518     [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 2, 4],
%e A248518     [1, 1, 2, 2, 4], [2, 2, 2, 4], [1, 1, 1, 3, 4], [1, 2, 3, 4], [3, 3, 4],
%e A248518     [1, 1, 4, 4], [2, 4, 4], [1, 1, 1, 1, 1, 5], [1, 1, 1, 2, 5], [1, 2, 2, 5],
%e A248518     [1, 1, 3, 5], [2, 3, 5], [1, 4, 5], [5, 5], [1, 1, 1, 1, 6], [1, 1, 2, 6],
%e A248518     [2, 2, 6], [1, 3, 6], [4, 6], [1, 1, 1, 7], [1, 2, 7], [3, 7], [1, 1, 8],
%e A248518     [2, 8], [1, 9], [10]]
%e A248518 If we excluse those that have a 1 in one of the parts, 11 partitions are left:
%e A248518 [[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]].
%e A248518 So a(10) = 11. - _N. J. A. Sloane_, Jun 28 2017
%e A248518 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;
%e A248518 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.
%t A248518 Table[Length[Select[IntegerPartitions[n],!MemberQ[Flatten[ IntegerDigits/@#],1]&]],{n,0,60}] (* _Harvey P. Dale_, Jun 28 2017 *)
%o A248518 (Haskell)
%o A248518 a248518 = p $ tail a052383_list where
%o A248518    p _          0 = 1
%o A248518    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%Y A248518 Cf. A052383, A248519.
%K A248518 nonn,base
%O A248518 0,5
%A A248518 _Reinhard Zumkeller_, Oct 07 2014