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A220412
Triangle read by rows, the coefficients of J. L. Fields generalized Bernoulli polynomials.
6
1, 0, 1, 0, 1, 5, 0, 4, 21, 35, 0, 18, 101, 210, 175, 0, 48, 286, 671, 770, 385, 0, 33168, 207974, 531531, 715715, 525525, 175175, 0, 8640, 56568, 154466, 231231, 205205, 105105, 25025, 0, 1562544, 10615548, 30582796, 49534277, 49689640, 31481450, 11911900
OFFSET
0,6
COMMENTS
The Fields polynomials are defined: F_{2*n}(x) = sum(k=0..n, x^k*T(n,k)/A220411(n)) and F_{2*n+1}(x) = 0. See A220002 for an application to the asymptotic expansion of the Catalan numbers.
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:
with(PolynomialTools): A220412_row := n -> CoefficientList(FieldsPoly( 2*i, x), x): seq(A220412_row(i), i=0..8);
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
Cf. A220411.
Sequence in context: A271951 A157700 A099645 * A199092 A167260 A137520
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 30 2012
STATUS
approved