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A260327 Triangle read by rows: T(n,k) (0 <= k <= n) gives numerators of coefficients in Nörlund's polynomials D_{2n}(x).

%I A260327 #44 Jul 04 2019 13:37:33
%S A260327 1,0,-1,0,2,5,0,-16,-42,-35,0,144,404,420,175,0,-768,-2288,-2684,
%T A260327 -1540,-385,0,1061376,3327584,4252248,2862860,1051050,175175,0,
%U A260327 -552960,-1810176,-2471456,-1849848,-820820,-210210,-25025,0,200005632,679395072,978649472,792548432,397517120,125925800,23823800,2127125
%N A260327 Triangle read by rows: T(n,k) (0 <= k <= n) gives numerators of coefficients in Nörlund's polynomials D_{2n}(x).
%H A260327 Jean-François Alcover, <a href="/A260327/b260327.txt">Table of n, a(n) for n = 0..230</a>
%H A260327 Max Alekseyev, <a href="https://mathoverflow.net/a/279368">An explicit representation for polynomials generated by a power of x/sin(x). Answer.</a> MathOverflow 2017.
%H A260327 N. E. Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen ueber Differenzenrechnung</a> Springer 1924, p. 460.
%H A260327 N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924; page 460 [Annotated scanned copy of pages 144-151 and 456-463]
%F A260327 E.g.f. Sum_{n>=0} D_{2n}(x) y^(2n)/(2n)! = (y/sinh(y))^x. - _Max Alekseyev_, Jul 04 2019
%e A260327 Triangle begins:
%e A260327 1,
%e A260327 0,-1,
%e A260327 0,2,5,
%e A260327 0,-16,-42,-35,
%e A260327 0,144,404,420,175,
%e A260327 0,-768,-2288,-2684,-1540,-385,
%e A260327 0,1061376,3327584,4252248,2862860,1051050,175175,
%e A260327 ...
%e A260327 The first few polynomials are (as listed in Nörlund, page 460):
%e A260327 [ 0] 1;
%e A260327 [ 2] -n/3;
%e A260327 [ 4]  n*(5*n + 2)/15;
%e A260327 [ 6] -n*(35*n^2  + 42*n     + 16)/63;
%e A260327 [ 8]  n*(175*n^3 + 420*n^2  + 404*n    + 144)/135;
%e A260327 [10] -n*(385*n^4 + 1540*n^3 + 2684*n^2 + 2288*n + 768)/99;
%p A260327 NorlundD := proc(n) if irem(n, 2) = 1 then return unapply(0, x) fi;
%p A260327 series((z/sin(z))^x, z, n+1): return unapply((-1)^iquo(n,2)*n!*coeff(%, z, n), x) end: A260327_row := n -> seq(coeff(numer(NorlundD(2*n)(x)),x,k), k=0..n):
%p A260327 for n from 0 to 6 do A260327_row(n) od; # _Peter Luschny_, Jul 01 2019
%t A260327 NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2] // Simplify;
%t A260327 Table[NorlundD[nu, n] // Together // Numerator // CoefficientList[#, n]&, {nu, 0, 12, 2}] (* _Jean-François Alcover_, Jul 01 2019 *)
%o A260327 (PARI) { A260327_row(n) = my(t,Y); Y=y+O(y^(2*n+2)); t = (2*n)! * Pol( polcoeff( exp( x * log(Y/sinh(Y)) + O(x^(n+1)) ), 2*n, y ) ); Vecrev(t*denominator(content(t))); } \\ _Max Alekseyev_, Jul 04 2019
%Y A260327 For denominators see A260326.
%K A260327 sign,tabl,look,frac
%O A260327 0,5
%A A260327 _N. J. A. Sloane_, Jul 25 2015
%E A260327 Typo in data and example corrected by _Jean-François Alcover_, Jul 01 2019
%E A260327 More terms by _Peter Luschny_, Jul 01 2019