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A238351 Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).

%I A238351 #21 Mar 03 2021 21:50:35
%S A238351 1,0,1,2,3,6,11,22,42,82,161,316,624,1235,2449,4864,9676,19267,38399,
%T A238351 76582,152819,305085,609282,1217140,2431992,4860306,9714696,19419870,
%U A238351 38824406,77624110,155208405,310352615,620601689,1241036325,2481803050,4963170896
%N A238351 Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).
%C A238351 Column k=0 of A238349 and of A238350.
%D A238351 M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences
%H A238351 Joerg Arndt and Alois P. Heinz, <a href="/A238351/b238351.txt">Table of n, a(n) for n = 0..1000</a>
%H A238351 M. Archibald, A. Blecher, and A. Knopfmacher, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Blecher/arch14.html">Fixed Points in Compositions and Words</a>, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
%F A238351 a(n) ~ c * 2^n, where c = A048651/2 = 0.14439404754330121... - _Vaclav Kotesovec_, May 01 2014
%e A238351 The a(7) = 22 such compositions are:
%e A238351 01:  [ 2 1 1 1 1 1 ]
%e A238351 02:  [ 2 1 1 1 2 ]
%e A238351 03:  [ 2 1 1 2 1 ]
%e A238351 04:  [ 2 1 1 3 ]
%e A238351 05:  [ 2 1 2 1 1 ]
%e A238351 06:  [ 2 1 2 2 ]
%e A238351 07:  [ 2 1 4 ]
%e A238351 08:  [ 2 3 1 1 ]
%e A238351 09:  [ 2 3 2 ]
%e A238351 10:  [ 2 4 1 ]
%e A238351 11:  [ 2 5 ]
%e A238351 12:  [ 3 1 1 1 1 ]
%e A238351 13:  [ 3 1 1 2 ]
%e A238351 14:  [ 3 1 2 1 ]
%e A238351 15:  [ 3 3 1 ]
%e A238351 16:  [ 3 4 ]
%e A238351 17:  [ 4 1 1 1 ]
%e A238351 18:  [ 4 1 2 ]
%e A238351 19:  [ 4 3 ]
%e A238351 20:  [ 5 1 1 ]
%e A238351 21:  [ 6 1 ]
%e A238351 22:  [ 7 ]
%p A238351 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238351        add(`if`(i=j, 0, b(n-j, i+1)), j=1..n))
%p A238351     end:
%p A238351 a:= n-> b(n, 1):
%p A238351 seq(a(n), n=0..50);
%t A238351 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, i+1]], {j, 1, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Nov 06 2014, after Maple *)
%K A238351 nonn
%O A238351 0,4
%A A238351 _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014