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

1, 1, -2, -1, -6, 0, 10, 16, 4, -11, -17, -12, -5, -1, -24, 84, -60, 30, -144, -48, 104, 186, 268, -12, -240, -436, -348, -46, 262, 444, 391, 199, -23, -166, -207, -172, -109, -55, -21, -6, -1, 120, -1200, 4560, -7740, 5064, -2472, 9768, -19152, 35004, -39408

Offset: 0

Author

John Tyler Rascoe, Sep 10 2024

Keywords

• sign
• tabf

Examples

			For row n = 2, C({1,2},x) = (-2*x^3 - x^4)/(1 + x + 2*x^2 - x^3 - x^4).
Triangle begins
  k=0  1  2  3   4   5   6   7   8   9  10   11   12   13  14  15
n=0 1;
n=1 .  1;
n=2 .  .  . -2, -1;
n=3 .  .  .  .   .   .  -6,  0, 10, 16,  4, -11, -17, -12, -5, -1;
		

Links

• John Tyler Rascoe, Rows n = 0..7, flattened

Crossrefs

Cf. A107429, A175669, A196847, A231147, A373196.

Programs

PARI

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)}
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}

Formula

C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) with C({},x) = 1.