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A276996 Numerators 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.

%I A276996 #22 Sep 09 2018 11:15:54
%S A276996 1,0,0,1,-1,1,0,1,-3,1,1,-1,6,-10,5,0,-1,-15,95,-40,16,239,-1,13,-85,
%T A276996 240,-237,79,0,403,21,385,-1575,3577,-2947,421,-46409,-239,3841,175,
%U A276996 861,-8036,45458,-10692,2673,0,-82451,-2657,56177,1638,19488,-85260,139656,-86472,19216
%N A276996 Numerators 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 A276996 The polynomials appear in certain asymptotic series for the Gamma function, cf. for example A181855/A181856 and A277000/A277001.
%H A276996 W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory (2016).
%F A276996 T(n,k) = Numerator([x^k] p_n(x)) where p_n(x) = Y_{n}(z_1, z_2, z_3,..., z_n) are the complete Bell polynomials evaluated at z_1 = 0 and z_k = (k-2)!*B_k(x) for k>1 and B_k(x) the Bernoulli polynomials.
%e A276996 Polynomials start:
%e A276996 p_0(x) = 1;
%e A276996 p_1(x) = 0;
%e A276996 p_2(x) = 1/6 + -x + x^2;
%e A276996 p_3(x) = (1/2)*x + -(3/2)*x^2 + x^3;
%e A276996 p_4(x) = 1/60 + -x + 6*x^2 + -10*x^3 + 5*x^4;
%e A276996 p_5(x) = -(1/6)*x + -(15/2)*x^2 + (95/3)*x^3 + -40*x^4 + 16*x^5;
%e A276996 p_6(x) = 239/504 + -(1/4)*x + (13/4)*x^2 + -85*x^3 + 240*x^4 + -237*x^5 + 79*x^6;
%e A276996 Triangle starts:
%e A276996 1;
%e A276996 0,   0;
%e A276996 1,  -1,   1;
%e A276996 0,   1,  -3,   1;
%e A276996 1,  -1,   6, -10,  5;
%e A276996 0,  -1, -15,  95, -40,   16;
%e A276996 239,-1,  13, -85, 240, -237, 79;
%p A276996 A276996_row := proc(n) local p;
%p A276996 p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)):
%p A276996 seq(numer(coeff(p(n,x),x,k)), k=0..n) end:
%p A276996 seq(A276996_row(n), n=0..9);
%p A276996 # Recurrence for the polynomials:
%p A276996 A276996_poly := proc(n,x) option remember; local z;
%p A276996 if n = 0 then return 1 fi; z := proc(k) option remember;
%p A276996 if k=1 then 0 else (k-2)!*bernoulli(k,x) fi end;
%p A276996 expand(add(binomial(n-1,j)*z(n-j)*A276996_poly(j,x),j=0..n-1)) end:
%p A276996 for n from 0 to 5 do sort(A276996_poly(n,x)) od;
%t A276996 CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
%t A276996 p[n_, x_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, x], {k, 2, n}]]];
%t A276996 row[0] = {1}; row[1] = {0, 0}; row[n_] := CoefficientList[p[n, x], x] // Numerator;
%t A276996 Table[row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Sep 09 2018 *)
%Y A276996 Cf. A276997 (denominators); T(2n,0) = A181855(n), T(n,n) = A203852(n).
%Y A276996 Cf. A276998.
%K A276996 sign,frac,tabl
%O A276996 0,9
%A A276996 _Peter Luschny_, Oct 01 2016