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

Author

Peter Luschny, Dec 30 2012

Keywords

• nonn
• tabl

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.

Examples

			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)
		

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.

Crossrefs

Cf. A220411.

Programs

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 *)

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)

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.