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

0,9

Comments

The polynomials appear in certain asymptotic series for the Gamma function, cf. for example A181855/A181856 and A277000/A277001.

Links

Table of n, a(n) for n=0..54.

W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

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.

Example

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;

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;

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 *)

Crossrefs

Cf. A276997 (denominators); T(2n,0) = A181855(n), T(n,n) = A203852(n).

Cf. A276998.

Sequence in context: A124372 A126470 A179701 * A102480 A157964 A140670

Adjacent sequences: A276993 A276994 A276995 * A276997 A276998 A276999

Keyword

sign,frac,tabl

Author

Peter Luschny, Oct 01 2016

Status

approved