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

%I A370414 #12 Feb 18 2024 12:05:59
%S A370414 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,
%T A370414 1,275,79,-133,-21,35,7,-7,1,-2339,275,79,-133,-21,7,7,-2,1,-11813,
%U A370414 -2339,825,79,-399,-189,21,3,-9,1,14217,-11813,-2339,1375,395,-399,-63,15,15,-5,1
%N A370414 T(n, k) = numerator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.
%C A370414 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, ...
%C A370414 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.
%C A370414 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.
%H A370414 Niels Erik Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen über Differenzenrechnung</a>, Springer 1924.
%F A370414 T(n, k) = numerator( n! * [z^k] [t^n] (t / (exp(t) - 1))^(1/2)*exp(z*t) ).
%e A370414 The lists of rational coefficients start:
%e A370414   [0] [        1]
%e A370414   [1] [     -1/4,        1]
%e A370414   [2] [     1/48,     -1/2,         1]
%e A370414   [3] [     1/64,     1/16,      -3/4,       1]
%e A370414   [4] [  -3/1280,     1/16,       1/8,      -1,     1]
%e A370414   [5] [ -19/3072,   -3/256,      5/32,    5/24,  -5/4,    1]
%e A370414   [6] [ 79/86016,  -19/512,    -9/256,    5/16,  5/16, -3/2,    1]
%e A370414   [7] [275/49152, 79/12288, -133/1024, -21/256, 35/64, 7/16, -7/4, 1]
%p A370414 egf := (t/(exp(t) - 1))^(1/2)*exp(z*t):
%p A370414 ser := series(egf, t, 16): ct := n -> n!*coeff(ser, t, n):
%p A370414 seq(seq(numer(coeff(ct(n), z, k)), k = 0..n), n = 0..10);
%t A370414 Table[Numerator@CoefficientList[NorlundB[n, 1/2, x], x] , {n, 0, 10}] // Flatten
%Y A370414 Cf. A370415, A370416/A370417 (x=1), A241885/A242225 (x=0).
%K A370414 sign,tabl,frac
%O A370414 0,9
%A A370414 _Peter Luschny_, Feb 18 2024