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

Author

Peter Luschny, Feb 18 2024

Keywords

• sign
• tabl
• frac

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.

Examples

			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]
		

Links

• Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924.

Crossrefs

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

Programs

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

Formula

T(n, k) = numerator( n! * [z^k] [t^n] (t / (exp(t) - 1))^(1/2)*exp(z*t) ).