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.
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, 240, -237, 79, 0, 403, 21, 385, -1575, 3577, -2947, 421, -46409, -239, 3841, 175, 861, -8036, 45458, -10692, 2673, 0, -82451, -2657, 56177, 1638, 19488, -85260, 139656, -86472, 19216
Offset: 0
Examples
Polynomials start: p_0(x) = 1; p_1(x) = 0; p_2(x) = 1/6 + -x + x^2; p_3(x) = (1/2)*x + -(3/2)*x^2 + x^3; p_4(x) = 1/60 + -x + 6*x^2 + -10*x^3 + 5*x^4; p_5(x) = -(1/6)*x + -(15/2)*x^2 + (95/3)*x^3 + -40*x^4 + 16*x^5; p_6(x) = 239/504 + -(1/4)*x + (13/4)*x^2 + -85*x^3 + 240*x^4 + -237*x^5 + 79*x^6; Triangle starts: 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, 240, -237, 79;
Links
- W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
Programs
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Maple
A276996_row := proc(n) local p; p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)): seq(numer(coeff(p(n,x),x,k)), k=0..n) end: seq(A276996_row(n), n=0..9); # Recurrence for the polynomials: A276996_poly := proc(n,x) option remember; local z; if n = 0 then return 1 fi; z := proc(k) option remember; if k=1 then 0 else (k-2)!*bernoulli(k,x) fi end; expand(add(binomial(n-1,j)*z(n-j)*A276996_poly(j,x),j=0..n-1)) end: for n from 0 to 5 do sort(A276996_poly(n,x)) od;
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Mathematica
CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}]; p[n_, x_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, x], {k, 2, n}]]]; row[0] = {1}; row[1] = {0, 0}; row[n_] := CoefficientList[p[n, x], x] // Numerator; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)
Formula
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.
Comments