%I #57 Nov 24 2025 08:04:00

%S 1,2,5,14,39,108,298,824,2278,6298,17412,48138,133085,367934,1017212,

%T 2812240,7774875,21494854,59425870,164292072,454211016,1255737086,

%U 3471680722,9598001976,26535171097,73360612640,202816837599,560718730852,1550194248403,4285753401052

%N Number of compositions of n such that within in each maximal run of k's, up to k parts within that run can be marked.

%H Alois P. Heinz, <a href="/A387637/b387637.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1 - Sum_{j>0} b(j,x)/(1 + b(j,x))) where b(j,x) = x^j/(1-x^j) + Sum_{i=1..j} x^(j*i)/(1 - x^j)^(i+1).

%e In the marked composition (1',3,3,3',3,3',1,1) of n = 18 up to 3 3's in the maximal run of 3's can be marked and a single 1 can be marked in each maximal run of 1's.

%e a(3) = 14 counts: (1,1,1), (1',1,1), (1,1',1), (1,1,1'), (1,2), (1',2), (1,2'), (1',2'), (2,1), (2',1), (2,1'), (2',1'), (3), (3').

%o (PARI)

%o b(f) = {x^f/(1-x^f) + sum(i=1,f, x^(f*i)/(1 - x^f)^(i+1) )}

%o A_x(N) = {Vec( 1/( 1- sum(j=1,N, b(j)/(1+b(j)))) + O('x^(N+1)))}

%Y Cf. A011782, A301503, A389508, A389509, A389511.

%K nonn,easy

%O 0,2

%A _John Tyler Rascoe_, Nov 18 2025