OFFSET
0,9
COMMENTS
Nørlund polynomials N(a, n, x) are generalizations of the powers 1, x, x^2, ... as well as of the Bernoulli polynomials 1, x - 1/2, x^2 - x + 1/6, ...
Parameter a = 0 gives the first case and a = 1 the second case. Here, we consider the case a = 1/2. You can think of it as a kind of square root of the Bernoulli polynomials. We give the coefficients of these polynomials, this sequence for the numerators, and A370415 for the denominators.
LINKS
Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924.
FORMULA
T(n, k) = numerator( n! * [z^k] [t^n] (t / (exp(t) - 1))^(1/2)*exp(z*t) ).
EXAMPLE
The lists of rational coefficients start:
[0] [ 1]
[1] [ -1/4, 1]
[2] [ 1/48, -1/2, 1]
[3] [ 1/64, 1/16, -3/4, 1]
[4] [ -3/1280, 1/16, 1/8, -1, 1]
[5] [ -19/3072, -3/256, 5/32, 5/24, -5/4, 1]
[6] [ 79/86016, -19/512, -9/256, 5/16, 5/16, -3/2, 1]
[7] [275/49152, 79/12288, -133/1024, -21/256, 35/64, 7/16, -7/4, 1]
MAPLE
egf := (t/(exp(t) - 1))^(1/2)*exp(z*t):
ser := series(egf, t, 16): ct := n -> n!*coeff(ser, t, n):
seq(seq(numer(coeff(ct(n), z, k)), k = 0..n), n = 0..10);
MATHEMATICA
Table[Numerator@CoefficientList[NorlundB[n, 1/2, x], x] , {n, 0, 10}] // Flatten
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Feb 18 2024
STATUS
approved