A276996 -id:A276996 - cp's OEIS frontend

A276997 Denominators 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, 1, 1, 6, 1, 1, 1, 2, 2, 1, 60, 1, 1, 1, 1, 1, 6, 2, 3, 1, 1, 504, 4, 4, 1, 1, 1, 1, 1, 24, 8, 12, 2, 2, 2, 1, 2160, 18, 9, 3, 2, 1, 3, 1, 1, 1, 60, 4, 6, 1, 5, 1, 1, 1, 1, 3168, 48, 16, 6, 3, 2, 2, 1, 2, 1, 1, 1, 288, 32, 144, 12, 12, 4, 2, 1, 6, 2, 1

Offset: 0

Peter Luschny, Oct 01 2016

For formulas and references see A276996.

Compare T(n,0) with A220411.

			Triangle starts:
     1;
     1,  1;
     6,  1,  1;
     1,  2,  2,  1;
    60,  1,  1,  1, 1;
     1,  6,  2,  3, 1, 1;
   504,  4,  4,  1, 1, 1, 1;
     1, 24,  8, 12, 2, 2, 2, 1;
  2160, 18,  9,  3, 2, 1, 3, 1, 1;

Cf. A276996 (numerators), A220411.

A276997_row := proc(n) local p;
p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)):
seq(denom(coeff(p(n,x),x,k)), k=0..n) end:
seq(A276997_row(n), n=0..11);

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] = {1, 1}; row[n_] := CoefficientList[p[n, x], x] // Denominator;
Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)

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

A276997 Denominators 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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