A220412 Triangle read by rows, the coefficients of J. L. Fields generalized Bernoulli polynomials.

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

Table of n, a(n) for n=0..43.

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

Adjacent sequences: A220409 A220410 A220411 * A220413 A220414 A220415

Keyword

nonn,tabl

Author

Peter Luschny, Dec 30 2012

Status

approved