A370414 T(n, k) = numerator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

1, -1, 1, 1, -1, 1, 1, 1, -3, 1, -3, 1, 1, -1, 1, -19, -3, 5, 5, -5, 1, 79, -19, -9, 5, 5, -3, 1, 275, 79, -133, -21, 35, 7, -7, 1, -2339, 275, 79, -133, -21, 7, 7, -2, 1, -11813, -2339, 825, 79, -399, -189, 21, 3, -9, 1, 14217, -11813, -2339, 1375, 395, -399, -63, 15, 15, -5, 1

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.

We also give the values of these polynomials at the point x = 1, which are analogous to the Bernoulli numbers; A370416 for the numerators, and A370417 for the denominators.

Links

Table of n, a(n) for n=0..65.

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

Cf. A370415, A370416/A370417 (x=1), A241885/A242225 (x=0).

Sequence in context: A063065 A324724 A051718 * A016472 A095346 A386236

Adjacent sequences: A370411 A370412 A370413 * A370415 A370416 A370417

Keyword

sign,tabl,frac

Author

Peter Luschny, Feb 18 2024

Status

approved