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A240736 Number of compositions of n having exactly one fixed point.

%I A240736 #20 Mar 03 2021 21:50:43
%S A240736 1,1,1,4,7,16,29,60,120,238,479,956,1910,3817,7633,15252,30491,60955,
%T A240736 121865,243650,487165,974112,1947851,3895086,7789153,15576624,
%U A240736 31150481,62296424,124585395,249158607,498297297,996562085,1993071152,3986055928,7971971230
%N A240736 Number of compositions of n having exactly one fixed point.
%D A240736 M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.
%H A240736 Joerg Arndt and Alois P. Heinz, <a href="/A240736/b240736.txt">Table of n, a(n) for n = 1..1000</a>
%H A240736 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 A240736 a(n) ~ c * 2^n, where c = A065442 * A048651 / 2 = 0.2319972162254452238942023675457837005318389885... - _Vaclav Kotesovec_, Sep 06 2014
%e A240736 a(4) = 4: 13, 22, 112, 1111.
%e A240736 a(5) = 7: 14, 32, 131, 221, 1112, 1121, 11111.
%p A240736 b:= proc(n, i) option remember; `if`(n=0, 1, series(
%p A240736       add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 2))
%p A240736     end:
%p A240736 a:= n-> coeff(b(n, 1), x, 1):
%p A240736 seq(a(n), n=1..40);
%t A240736 b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 2}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 1}]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Nov 06 2014, after Maple *)
%Y A240736 Column k=1 of A238349 and of A238350.
%K A240736 nonn
%O A240736 1,4
%A A240736 _Joerg Arndt_ and _Alois P. Heinz_, Apr 11 2014