A276998 Coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x) where B_k(x) are the Bernoulli polynomials.

1, 1, 2, 1, 12, 6, -1, 72, 24, -24, 1, 1440, 120, -960, 200, 37, 43200, -9360, -44280, 20640, 3750, -1493, 1814400, -997920, -2484720, 2028600, 271740, -378966, 14017, 25401600, -23042880, -42497280, 54159840, 3328080, -18236064, 1977248, 751267

Offset: 0

Author

Peter Luschny, Oct 03 2016

Keywords

• sign
• tabf

Examples

			Sequence of rational polynomials P_n(x) starts:
1;
1;
(2*x + 1)/2;
(12*x^2 + 6*x - 1)/6;
(72*x^3 + 24*x^2 - 24*x + 1)/12;
(1440*x^4 + 120*x^3 - 960*x^2 + 200*x + 37)/60;
(43200*x^5 - 9360*x^4 - 44280*x^3 + 20640*x^2 + 3750*x - 1493)/360;
Triangle starts:
[1]
[1]
[2, 1]
[12, 6, -1]
[72, 24, -24, 1]
[1440, 120, -960, 200, 37]
[43200, -9360, -44280, 20640, 3750, -1493]
		

Crossrefs

T(n,0) = A277174(n)/n for n>=1.

Cf. A048803, A276996, A276997.

Programs

Maple

P := proc(n) local B;
B := (n, x) -> CompleteBellB(n, seq(k!*bernoulli(k, x), k=0..n)):
sort(A048803(n)*B(n, x)) end:
A276998_row := n -> PolynomialTools[CoefficientList](P(n), x, termorder=reverse):
seq(op(A276998_row(n)), n=0..8);
# Recurrence for the rational polynomials:
A276998_poly := proc(n,x) option remember; local z; if n = 0 then return 1 fi;
z := proc(k) option remember; k!*bernoulli(k,x) end;
expand(add(binomial(n-1,j)*z(n-j-1)*A276998_poly(j,x),j=0..n-1)) end:
for n from 0 to 5 do sort(A276998_poly(n,x)) od;

Mathematica

(* b = A048803 *) b[0] = 1; b[n_] := b[n] = b[n-1] First @ Select[ Reverse @ Divisors[n], SquareFreeQ, 1];
CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
B[n_, x_] := CompleteBellB[n, Table[k!*BernoulliB[k, x], {k, 0, n}]];
P[n_] := b[n] B[n, x];
row[0] = {1}; row[n_] := CoefficientList[P[n], x] // Reverse;
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)

Formula

P_n(x) = Y_n(x_0, x_1, x_2,..., x_n), the complete Bell polynomials evaluated at x_k = k!*B_k(x) and B_k(x) the Bernoulli polynomials.

T(n,k) = A048803(n)*[x^k] P_n(x).