A220412 - OEIS

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

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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)