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A383253 Number of compositions of n with parts in standard order.

%I A383253 #40 May 08 2025 04:17:09
%S A383253 1,1,1,2,3,5,9,16,29,53,98,182,340,638,1202,2273,4312,8204,15650,
%T A383253 29925,57344,110101,211771,407987,787174,1520851,2942030,5697842,
%U A383253 11046881,21438881,41645541,80967881,157547508,306791828,597847686,1165828440,2274890125
%N A383253 Number of compositions of n with parts in standard order.
%C A383253 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 A383253 Alois P. Heinz, <a href="/A383253/b383253.txt">Table of n, a(n) for n = 0..3333</a>
%F A383253 G.f.: 1 + Sum_{i>0} x^(i*(i+1)/2) / Product_{j=1..i} 1 - (x - x^(j+1))/(1 - x).
%e A383253 a(6) = 9 counts: (1,1,1,1,1,1), (1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,2,1,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,1), (1,2,3).
%p A383253 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A383253       b(n-j, max(i, j)), j=1..min(n, i+1)))
%p A383253     end:
%p A383253 a:= n-> b(n, 0):
%p A383253 seq(a(n), n=0..36);  # _Alois P. Heinz_, May 08 2025
%o A383253 (PARI)
%o A383253 A_x(N) = {my(x='x+O('x^(N+1))); Vec(1 + sum(i=1,(N/2)+1, x^(i*(i+1)/2)/prod(j=1,i, 1 - (x-x^(j+1))/(1-x))))}
%o A383253 A_x(40)
%Y A383253 Cf. A000110, A011782, A047998, A107429, A126347, A278984, (row sums of A383713).
%K A383253 nonn,easy
%O A383253 0,4
%A A383253 _John Tyler Rascoe_, May 06 2025