cp's OEIS Frontend

A373420 Number of Carlitz compositions of n (see A003242) such that the first and last parts are equal.

%I A373420 #27 Aug 16 2024 20:52:34
%S A373420 1,1,1,1,2,3,2,7,11,17,26,54,86,155,272,464,816,1447,2507,4400,7706,
%T A373420 13456,23570,41293,72212,126394,221282,387219,677714,1186311,2076170,
%U A373420 3633761,6360219,11131698,19483066,34100455,59683664,104460655,182832044,319999739
%N A373420 Number of Carlitz compositions of n (see A003242) such that the first and last parts are equal.
%F A373420 G.f.: 1 + Sum_{i>0} (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2 where C(x) is the g.f. for A003242.
%e A373420 a(7) = 7 counts: (1,2,1,2,1), (1,2,3,1), (1,3,2,1), (1,5,1), (2,3,2), (3,1,3), and (7).
%o A373420 (PARI)
%o A373420 C_x(N) = {my(g=1/(1-sum(k=1, N, x^k/(1+x^k))));g}
%o A373420 A_x(i,N) = {my( x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
%o A373420 D_x(N) = {my( x='x+O('x^N), f=1+sum(i=1,N, A_x(i,N))); Vec(f)}
%o A373420 D_x(40)
%Y A373420 Cf. A001045, A003242, A285981, A291941, A374517.
%K A373420 nonn,easy
%O A373420 0,5
%A A373420 _John Tyler Rascoe_, Aug 16 2024