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A376117 Irregular triangle of numerator polynomial coefficients of C({1..n},x), T(n,k) for n >= 0 and k >= A000217(n).

%I A376117 #10 Oct 16 2024 22:28:52
%S A376117 1,1,-2,-1,-6,0,10,16,4,-11,-17,-12,-5,-1,-24,84,-60,30,-144,-48,104,
%T A376117 186,268,-12,-240,-436,-348,-46,262,444,391,199,-23,-166,-207,-172,
%U A376117 -109,-55,-21,-6,-1,120,-1200,4560,-7740,5064,-2472,9768,-19152,35004,-39408
%N A376117 Irregular triangle of numerator polynomial coefficients of C({1..n},x), T(n,k) for n >= 0 and k >= A000217(n).
%H A376117 John Tyler Rascoe, <a href="/A376117/b376117.txt">Rows n = 0..7, flattened</a>
%F A376117 C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) with C({},x) = 1.
%e A376117 For row n = 2, C({1,2},x) = (-2*x^3 - x^4)/(1 + x + 2*x^2 - x^3 - x^4).
%e A376117 Triangle begins
%e A376117   k=0  1  2  3   4   5   6   7   8   9  10   11   12   13  14  15
%e A376117 n=0 1;
%e A376117 n=1 .  1;
%e A376117 n=2 .  .  . -2, -1;
%e A376117 n=3 .  .  .  .   .   .  -6,  0, 10, 16,  4, -11, -17, -12, -5, -1;
%o A376117 (PARI)
%o A376117 C_x(s)={my( g=if(#s <1, 1, sum(i=1, #s, C_x(s[^i]) * x^(s[i]) )/(1-sum(i=1, #s, x^(s[i]))))); return(g)}
%o A376117 A376117_row(n)={my(t=n*(n+1)/2, c=C_x([1..n]), d=poldegree(numerator(c))-t, z=vector(d+1)); for(k=0,d,z[k+1]=polcoeff(numerator(c),k+t)); z}
%Y A376117 Cf. A107429, A175669, A196847, A231147, A373196.
%K A376117 sign,tabf
%O A376117 0,3
%A A376117 _John Tyler Rascoe_, Sep 10 2024