A387126
Triangle read by rows: T(n, k) = (n! / (n - k)!) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.
Original entry on oeis.org
1, 1, 1, 2, 4, 4, 6, 18, 36, 36, 12, 48, 144, 288, 288, 60, 300, 1200, 3600, 7200, 7200, 360, 2160, 10800, 43200, 129600, 259200, 259200, 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800, 5040, 40320, 282240, 1693440, 8467200, 33868800, 101606400, 203212800, 203212800
Offset: 0
Triangle begins:
[0] 1;
[1] 1, 1;
[2] 2, 4, 4;
[3] 6, 18, 36, 36;
[4] 12, 48, 144, 288, 288;
[5] 60, 300, 1200, 3600, 7200, 7200;
[6] 360, 2160, 10800, 43200, 129600, 259200, 259200;
[7] 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800;
-
A387126 := (n, k) -> mul(NumberTheory:-Radical(j), j = 1..n) * n! / (n - k)!:
-
A387126[n_, k_] := Pochhammer[n-k+1, k] Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
Table[A387126[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
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
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]
-
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 *)
Showing 1-2 of 2 results.