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Denominators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.
2

%I #17 Sep 09 2018 11:15:33

%S 1,1,1,6,1,1,1,2,2,1,60,1,1,1,1,1,6,2,3,1,1,504,4,4,1,1,1,1,1,24,8,12,

%T 2,2,2,1,2160,18,9,3,2,1,3,1,1,1,60,4,6,1,5,1,1,1,1,3168,48,16,6,3,2,

%U 2,1,2,1,1,1,288,32,144,12,12,4,2,1,6,2,1

%N Denominators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.

%C For formulas and references see A276996.

%C Compare T(n,0) with A220411.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 6, 1, 1;

%e 1, 2, 2, 1;

%e 60, 1, 1, 1, 1;

%e 1, 6, 2, 3, 1, 1;

%e 504, 4, 4, 1, 1, 1, 1;

%e 1, 24, 8, 12, 2, 2, 2, 1;

%e 2160, 18, 9, 3, 2, 1, 3, 1, 1;

%p A276997_row := proc(n) local p;

%p p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)):

%p seq(denom(coeff(p(n,x),x,k)), k=0..n) end:

%p seq(A276997_row(n), n=0..11);

%t CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];

%t p[n_, x_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, x], {k, 2, n}]]];

%t row[0] = {1}; row[1] = {1, 1}; row[n_] := CoefficientList[p[n, x], x] // Denominator;

%t Table[row[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Sep 09 2018 *)

%Y Cf. A276996 (numerators), A220411.

%K nonn,frac,tabl

%O 0,4

%A _Peter Luschny_, Oct 01 2016