OFFSET
0,6
COMMENTS
REFERENCES
Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969, page 34.
LINKS
J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
FORMULA
See Y. L. Luke 2.8(3) for the generalized Bernoulli polynomials and 2.11(16) for the special case of Fields polynomials. The Maple and Sage programs give a recursive definition.
EXAMPLE
The coefficients T(n,k):
[0], [1]
[1], [0, 1]
[2], [0, 1, 5]
[3], [0, 4, 21, 35]
[4], [0, 18, 101, 210, 175]
[5], [0, 48, 286, 671, 770, 385]
The Fields polynomials:
F_0 (x) = 1 / 1
F_2 (x) = x / (-6)
F_4 (x) = (5*x^2+x) / 60
F_6 (x) = (35*x^3+21*x^2+4*x) / (-504)
F_8 (x) = (175*x^4+210*x^3+101*x^2+18*x) / 2160
F_10(x) = (385*x^5+770*x^4+671*x^3+286*x^2+48*x) / (-3168)
MAPLE
FieldsPoly := proc(n, x) local recP, P; recP := proc(n, x) option remember; local k; if n = 0 then return(1) fi; -2*x*add(binomial(n-1, 2*k+1)* bernoulli(2*k+2)/(2*k+2)*recP(n-2*k-2, x), k=0..(n/2-1)) end:
P := recP(n, x); (-1)^iquo(n, 2)*denom(P); sort(expand(P*%)) end:
MATHEMATICA
F[0, _] = 1; F[n_, x_] := F[n, x] = -2*x*Sum[Binomial[n-1, 2*k+1]*BernoulliB[2*k+2]/(2*k+2)*F[n-2*k-2, x], {k, 0, n/2-1}]; t[n_, k_] := Coefficient[(-1)^Mod[n, 2]*F[2*n, x] // Together // Numerator, x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)
PROG
(Sage)
@CachedFunction
def FieldsPoly(n):
A = PolynomialRing(QQ, 'x')
x = A.gen()
if n == 0: return A(1)
return -2*x*add(binomial(n-1, 2*k+1)*bernoulli(2*k+2)/(2*k+2)*FieldsPoly(n-2*k-2) for k in (0..n-1))
def FieldsCoeffs(n):
P = FieldsPoly(n)
d = (-1)^(n//2) * denominator(P)
return list(d * P)
A220412_row = lambda n: FieldsCoeffs(2*n)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 30 2012
STATUS
approved