A260327 Triangle read by rows: T(n,k) (0 <= k <= n) gives numerators of coefficients in Nörlund's polynomials D_{2n}(x).
1, 0, -1, 0, 2, 5, 0, -16, -42, -35, 0, 144, 404, 420, 175, 0, -768, -2288, -2684, -1540, -385, 0, 1061376, 3327584, 4252248, 2862860, 1051050, 175175, 0, -552960, -1810176, -2471456, -1849848, -820820, -210210, -25025, 0, 200005632, 679395072, 978649472, 792548432, 397517120, 125925800, 23823800, 2127125
Offset: 0
Examples
Triangle begins: 1, 0,-1, 0,2,5, 0,-16,-42,-35, 0,144,404,420,175, 0,-768,-2288,-2684,-1540,-385, 0,1061376,3327584,4252248,2862860,1051050,175175, ... The first few polynomials are (as listed in Nörlund, page 460): [ 0] 1; [ 2] -n/3; [ 4] n*(5*n + 2)/15; [ 6] -n*(35*n^2 + 42*n + 16)/63; [ 8] n*(175*n^3 + 420*n^2 + 404*n + 144)/135; [10] -n*(385*n^4 + 1540*n^3 + 2684*n^2 + 2288*n + 768)/99;
Links
- Jean-François Alcover, Table of n, a(n) for n = 0..230
- Max Alekseyev, An explicit representation for polynomials generated by a power of x/sin(x). Answer. MathOverflow 2017.
- N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 460.
- N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 460 [Annotated scanned copy of pages 144-151 and 456-463]
Crossrefs
For denominators see A260326.
Programs
-
Maple
NorlundD := proc(n) if irem(n, 2) = 1 then return unapply(0, x) fi; 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): for n from 0 to 6 do A260327_row(n) od; # Peter Luschny, Jul 01 2019
-
Mathematica
NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2] // Simplify; Table[NorlundD[nu, n] // Together // Numerator // CoefficientList[#, n]&, {nu, 0, 12, 2}] (* Jean-François Alcover, Jul 01 2019 *) -
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
Formula
E.g.f. Sum_{n>=0} D_{2n}(x) y^(2n)/(2n)! = (y/sinh(y))^x. - Max Alekseyev, Jul 04 2019
Extensions
Typo in data and example corrected by Jean-François Alcover, Jul 01 2019
More terms by Peter Luschny, Jul 01 2019