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A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

%I A372888 #13 May 24 2024 14:37:10
%S A372888 0,1,2,7,13,31,66,138,279,581,1173,2375,4783,9630,19316,38802,77689,
%T A372888 155673,311639,623845,1248179,2497719,4996387,9995304,19992908,
%U A372888 39990902,79986136,159983241,319975073,639971495,1279962115,2559966847,5119970499,10240030209
%N A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).
%H A372888 Alois P. Heinz, <a href="/A372888/b372888.txt">Table of n, a(n) for n = 0..3321</a>
%F A372888 a(n) = Sum_{k=1..n} 2^(k-1) * A015716(n,k). - _Alois P. Heinz_, May 24 2024
%e A372888 The strict partitions of 6 are (6), (5,1), (4,2), (3,2,1), with respective binary ranks 32, 17, 10, 7 with sum 66, so a(6) = 66.
%p A372888 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A372888       `if`(n=0, [1,0], b(n, i-1)+ (p-> [0, p[1]*2^(i-1)]
%p A372888           +p)(b(n-i, min(n-i, i-1)))))
%p A372888     end:
%p A372888 a:= n-> b(n$2)[2]:
%p A372888 seq(a(n), n=0..33);  # _Alois P. Heinz_, May 23 2024
%t A372888 Table[Total[Total[2^(#-1)]& /@ Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]
%Y A372888 Row sums of A118462 (binary ranks of strict partitions).
%Y A372888 For Heinz number the non-strict version is A145519, row sums of A215366.
%Y A372888 For Heinz number (not binary rank) we have A147655, row sums of A246867.
%Y A372888 The non-strict version is A372890.
%Y A372888 A000009 counts strict partitions, ranks A005117.
%Y A372888 A048675 gives binary rank of prime indices, distinct A087207.
%Y A372888 A277905 groups all positive integers by binary rank of prime indices.
%Y A372888 Binary indices (A048793):
%Y A372888 - length A000120, complement A023416
%Y A372888 - min A001511, opposite A000012
%Y A372888 - max A029837 or A070939, opposite A070940
%Y A372888 - sum A029931, product A096111
%Y A372888 - reverse A272020
%Y A372888 - complement A368494, sum A359400
%Y A372888 - opposite A371572, sum A230877
%Y A372888 - opposite complement A371571, sum A359359
%Y A372888 Cf. A000041, A005940, A015716, A018819, A019565, A118457, A225620, A344086.
%K A372888 nonn
%O A372888 0,3
%A A372888 _Gus Wiseman_, May 23 2024