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A383751 Number of Carlitz compositions of n with parts in standard order.

%I A383751 #12 May 09 2025 16:54:31
%S A383751 1,1,0,1,1,0,2,3,2,5,8,10,19,31,44,73,123,193,315,524,847,1392,2317,
%T A383751 3810,6303,10506,17451,29066,48603,81223,135965,228153,383014,643756,
%U A383751 1083693,1825640,3078574,5197246,8780823,14847669,25128385,42558687,72131730,122343844
%N A383751 Number of Carlitz compositions of n with parts in standard order.
%C A383751 A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared. All compositions of this kind have first part 1.
%H A383751 Alois P. Heinz, <a href="/A383751/b383751.txt">Table of n, a(n) for n = 0..4135</a>
%F A383751 G.f.: Sum_{i>=0} G(i) where G(k) = G(k-1) * x*k * (1 + 1/(1 - Sum_{j=1..k} ( x^j/(1+x^j) )) * Sum_{j=1..k-1} ( x^j/(1+x^j) )) and G(0) = 1.
%e A383751 a(9) = 5 counts: (1,2,1,2,1,2), (1,2,1,2,3), (1,2,1,3,2), (1,2,3,1,2), (1,2,3,2,1).
%p A383751 b:= proc(n, l, i) option remember; `if`(n=0, 1, add(
%p A383751       `if`(j=l, 0, b(n-j, j, max(i, j))), j=1..min(n, i+1)))
%p A383751     end:
%p A383751 a:= n-> b(n, 0$2):
%p A383751 seq(a(n), n=0..43);  # _Alois P. Heinz_, May 09 2025
%o A383751 (PARI)
%o A383751 G(k,N) = {my(x='x+O('x^N)); if(k<1,1, G(k-1,N) * x^k * (1 + (1/(1 - sum(j=1,k, x^j/(1+x^j)))) * sum(j=1,k-1, x^j/(1+x^j))))}
%o A383751 A_x(N) = {my(x='x+O('x^N)); Vec(sum(i=0,N/2+1, G(i,N+1)))}
%o A383751 A_x(50)
%Y A383751 Cf. A000110, A003242, A011782, A047998, A107429, A126347, A278984, (column k=0 of A380822), A383253, A383713.
%K A383751 nonn,easy
%O A383751 0,7
%A A383751 _John Tyler Rascoe_, May 08 2025