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A323583 Number of ways to split an integer partition of n into consecutive subsequences.

%I A323583 #13 Jan 01 2023 19:21:25
%S A323583 1,1,3,7,17,37,83,175,373,773,1603,3275,6693,13557,27447,55315,111397,
%T A323583 223769,449287,900795,1805465,3615929,7240327,14491623,29001625,
%U A323583 58027017,116093259,232237583,464558201,929224589,1858623819,3717475031,7435314013,14871103069
%N A323583 Number of ways to split an integer partition of n into consecutive subsequences.
%F A323583 a(n) = A070933(n)/2.
%F A323583 O.g.f.: (1/2)*Product_{n >= 1} 1/(1 - 2*x^n).
%F A323583 G.f.: 1 + Sum_{k>=1} 2^(k - 1) * x^k / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Jan 28 2020
%e A323583 The a(3) = 7 ways to split an integer partition of 3 into consecutive subsequences are (3), (21), (2)(1), (111), (11)(1), (1)(11), (1)(1)(1).
%p A323583 b:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i<1, 0,
%p A323583       b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
%p A323583     end:
%p A323583 a:= n-> ceil(b(n$2)):
%p A323583 seq(a(n), n=0..33);  # _Alois P. Heinz_, Jan 01 2023
%t A323583 Table[Sum[2^(Length[ptn]-1),{ptn,IntegerPartitions[n]}],{n,40}]
%t A323583 (* Second program: *)
%t A323583 (1/2) CoefficientList[1 - 1/QPochhammer[2, x] + O[x]^100 , x] (* _Jean-François Alcover_, Jan 02 2022, after _Vladimir Reshetnikov_ in A070933 *)
%Y A323583 Cf. A006951, A070933, A100883, A279784, A279786, A323433, A323582.
%K A323583 nonn
%O A323583 0,3
%A A323583 _Gus Wiseman_, Jan 19 2019