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

%I A276998 #15 Sep 09 2018 11:16:10
%S A276998 1,1,2,1,12,6,-1,72,24,-24,1,1440,120,-960,200,37,43200,-9360,-44280,
%T A276998 20640,3750,-1493,1814400,-997920,-2484720,2028600,271740,-378966,
%U A276998 14017,25401600,-23042880,-42497280,54159840,3328080,-18236064,1977248,751267
%N A276998 Coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x) where B_k(x) are the Bernoulli polynomials.
%F A276998 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.
%F A276998 T(n,k) = A048803(n)*[x^k] P_n(x).
%e A276998 Sequence of rational polynomials P_n(x) starts:
%e A276998 1;
%e A276998 1;
%e A276998 (2*x + 1)/2;
%e A276998 (12*x^2 + 6*x - 1)/6;
%e A276998 (72*x^3 + 24*x^2 - 24*x + 1)/12;
%e A276998 (1440*x^4 + 120*x^3 - 960*x^2 + 200*x + 37)/60;
%e A276998 (43200*x^5 - 9360*x^4 - 44280*x^3 + 20640*x^2 + 3750*x - 1493)/360;
%e A276998 Triangle starts:
%e A276998 [1]
%e A276998 [1]
%e A276998 [2, 1]
%e A276998 [12, 6, -1]
%e A276998 [72, 24, -24, 1]
%e A276998 [1440, 120, -960, 200, 37]
%e A276998 [43200, -9360, -44280, 20640, 3750, -1493]
%p A276998 P := proc(n) local B;
%p A276998 B := (n, x) -> CompleteBellB(n, seq(k!*bernoulli(k, x), k=0..n)):
%p A276998 sort(A048803(n)*B(n, x)) end:
%p A276998 A276998_row := n -> PolynomialTools[CoefficientList](P(n), x, termorder=reverse):
%p A276998 seq(op(A276998_row(n)), n=0..8);
%p A276998 # Recurrence for the rational polynomials:
%p A276998 A276998_poly := proc(n,x) option remember; local z; if n = 0 then return 1 fi;
%p A276998 z := proc(k) option remember; k!*bernoulli(k,x) end;
%p A276998 expand(add(binomial(n-1,j)*z(n-j-1)*A276998_poly(j,x),j=0..n-1)) end:
%p A276998 for n from 0 to 5 do sort(A276998_poly(n,x)) od;
%t A276998 (* b = A048803 *) b[0] = 1; b[n_] := b[n] = b[n-1] First @ Select[ Reverse @ Divisors[n], SquareFreeQ, 1];
%t A276998 CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
%t A276998 B[n_, x_] := CompleteBellB[n, Table[k!*BernoulliB[k, x], {k, 0, n}]];
%t A276998 P[n_] := b[n] B[n, x];
%t A276998 row[0] = {1}; row[n_] := CoefficientList[P[n], x] // Reverse;
%t A276998 Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Sep 09 2018 *)
%Y A276998 T(n,0) = A277174(n)/n for n>=1.
%Y A276998 Cf. A048803, A276996, A276997.
%K A276998 sign,tabf
%O A276998 0,3
%A A276998 _Peter Luschny_, Oct 03 2016