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A370929 Number of compositions of n with parts (p_1, ..., p_i) such that the set of adjacent differences is a subset of {-k,k} for some k > 0 and the number of parts equals ceiling(p_1/k).

%I A370929 #16 Mar 09 2024 19:17:45
%S A370929 1,1,1,2,2,3,4,3,5,5,7,5,9,7,7,10,11,9,13,9,13,17,11,14,19,15,13,20,
%T A370929 19,18,23,19,20,26,21,20,32,22,25,27,33,25,37,21,34,36,35,24,50,26,40,
%U A370929 37,44,32,51,31,48,46,49,34,65,40,45,54,56,48,63,42,58
%N A370929 Number of compositions of n with parts (p_1, ..., p_i) such that the set of adjacent differences is a subset of {-k,k} for some k > 0 and the number of parts equals ceiling(p_1/k).
%H A370929 John Tyler Rascoe, <a href="/A370929/b370929.txt">Table of n, a(n) for n = 0..1000</a>
%F A370929 G.f.: 1 + Sum_{p>0} Sum_{k=1..p} x^(p*i) * Product_{j=1..i-1} (x^(-j*k) + x^(j*k)), where i = ceiling(p/k).
%e A370929 The compositions for n = 6 and n = 8 are:
%e A370929 6: [6], [5,1], [4,2], [3,2,1].
%e A370929 8: [8], [7,1], [6,2], [3,2,3], [3,5].
%o A370929 (PARI)
%o A370929 { my(N=75, x='x+O('x^N));
%o A370929 my(gf= 1 + sum(p=1, N, sum(k=1, p, x^(p*ceil(p/k)) * prod(j=1, ceil(p/k)-1, (x^(-j*k) + x^(j*k))))));
%o A370929 Vec(gf) }
%Y A370929 Compositions such that no adjacent parts are equal is A003242.
%Y A370929 Compositions such that the set of adjacent differences is a subset of {-1,1} is A173258 and {-2,2} is A214254.
%Y A370929 The array A214247 counts compositions such that the set of adjacent differences is a subset of {-k,k}.
%K A370929 nonn,easy
%O A370929 0,4
%A A370929 _John Tyler Rascoe_, Mar 06 2024