A276997 -id:A276997 - cp's OEIS frontend

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

Peter Luschny, Oct 01 2016

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

			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;

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

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

Cf. A276998.

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;

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 *)

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.

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

Original entry on oeis.org

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

Peter Luschny, Oct 03 2016

			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]

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

Cf. A048803, A276996, A276997.

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;

(* 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 *)

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).

cp's OEIS Frontend

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.

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