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A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

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%I A323776 #9 Jan 28 2019 10:25:39
%S A323776 1,3,7,16,40,119,450,2253,15207,139190,1731703,29335875,677864041,
%T A323776 21400069232,924419728471,54716596051100,4443400439075834,
%U A323776 495676372493566749,76041424515817042402,16060385520094706930608,4674665948889147697184915
%N A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).
%C A323776 Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.
%H A323776 Seiichi Manyama, <a href="/A323776/b323776.txt">Table of n, a(n) for n = 1..120</a>
%e A323776 The a(1) = 1 through a(4) = 16 partitions of partitions:
%e A323776   (1)  (2)     (4)           (8)
%e A323776        (11)    (22)          (44)
%e A323776        (1)(1)  (1111)        (2222)
%e A323776                (2)(2)        (4)(4)
%e A323776                (2)(11)       (4)(22)
%e A323776                (11)(11)      (22)(22)
%e A323776                (1)(1)(1)(1)  (4)(1111)
%e A323776                              (11111111)
%e A323776                              (22)(1111)
%e A323776                              (1111)(1111)
%e A323776                              (2)(2)(2)(2)
%e A323776                              (2)(2)(2)(11)
%e A323776                              (2)(2)(11)(11)
%e A323776                              (2)(11)(11)(11)
%e A323776                              (11)(11)(11)(11)
%e A323776                              (1)(1)(1)(1)(1)(1)(1)(1)
%t A323776 Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]
%o A323776 (PARI) a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ _Michel Marcus_, Jan 28 2019
%Y A323776 Cf. A000123, A001970, A002577, A006171, A007716, A034729,  A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.
%K A323776 nonn
%O A323776 1,2
%A A323776 _Gus Wiseman_, Jan 27 2019