OFFSET
0,9
COMMENTS
Consider a family of integrals I_m(n) = Integral_{x=0..1} P'(n, x)^m with P'(n,x) = Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!*x^k (see A278075 for the coefficients).
I_1(n) are the Bernoulli numbers A164555/A027642, I_2(n) are the Bernoulli median numbers A212196/A181131, I_3(n) are the numbers A291449/A291450.
The coefficients of the polynomials P_n(x)^m are for m = 1 A290694/A290695 and for m = 2 A291447/A291448.
Only omega(Clausen(n)) = A001221(A160014(n,1)) = A067513(n) coefficients are rational numbers if n is even. For odd n > 1 there are two rational coefficients.
Let C_k(n) = [x^k] P_n(x), k > 0 and n even. Conjecture: k is a prime factor of Clausen(n) <=> k = denominator(C_k(n)) <=> k does not divide Stirling2(n, k-1)*(k-1)!. (Note that by a comment in A019538 Stirling2(n, k-1)*(k-1)! is the number of chain topologies on an n-set having k open sets.)
LINKS
Peter Luschny, Illustrating A290694.
FORMULA
T(n, k) = Numerator(Stirling2(n, k - 1)*(k - 1)!/k) if k > 0 else 0; for n >= 0 and 0 <= k <= n+1.
EXAMPLE
Triangle starts:
[0, 1]
[0, 0, 1]
[0, 0, -1, 2]
[0, 0, 1, -2, 3]
[0, 0, -1, 14, -9, 24]
[0, 0, 1, -10, 75, -48, 20]
[0, 0, -1, 62, -135, 312, -300, 720]
The first few polynomials are:
P_0(x) = x.
P_1(x) = (1/2)*x^2.
P_2(x) = -(1/2)*x^2 + (2/3)*x^3.
P_3(x) = (1/2)*x^2 - 2*x^3 + (3/2)*x^4.
P_4(x) = -(1/2)*x^2 + (14/3)*x^3 - 9*x^4 + (24/5)*x^5.
P_5(x) = (1/2)*x^2 - 10*x^3 + (75/2)*x^4 - 48*x^5 + 20*x^6.
P_6(x) = -(1/2)*x^2 + (62/3)*x^3 - 135*x^4 + 312*x^5 - 300*x^6 + (720/7)*x^7.
Evaluated at x = 1 this gives an additive decomposition of the Bernoulli numbers:
B(0) = 1 = 1.
B(1) = 1/2 = 1/2.
B(2) = 1/6 = -1/2 + 2/3.
B(3) = 0 = 1/2 - 2 + 3/2.
B(4) = -1/30 = -1/2 + 14/3 - 9 + 24/5.
B(5) = 0 = 1/2 - 10 + 75/2 - 48 + 20.
B(6) = 1/42 = -1/2 + 62/3 - 135 + 312 - 300 + 720/7.
MAPLE
BG_row := proc(m, n, frac, val) local F, g, v;
F := (n, x) -> add((-1)^(n-k)*Stirling2(n, k)*k!*x^k, k=0..n):
g := x -> int(F(n, x)^m, x):
`if`(val = "val", subs(x=1, g(x)), [seq(coeff(g(x), x, j), j=0..m*n+1)]):
`if`(frac = "num", numer(%), denom(%)) end:
seq(BG_row(1, n, "num", "val"), n=0..16); # A164555
seq(BG_row(1, n, "den", "val"), n=0..16); # A027642
seq(print(BG_row(1, n, "num", "poly")), n=0..12); # A290694 (this seq.)
seq(print(BG_row(1, n, "den", "poly")), n=0..12); # A290695
# Alternatively:
T_row := n -> numer(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 6 do T_row(n) od;
MATHEMATICA
T[n_, k_] := If[k > 0, Numerator[StirlingS2[n, k - 1]*(k - 1)! / k], 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n+1}] // Flatten
CROSSREFS
KEYWORD
sign,tabf,frac
AUTHOR
Peter Luschny, Aug 24 2017
STATUS
approved