# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a260327 Showing 1-1 of 1 %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, Table of n, a(n) for n = 0..230 %H A260327 Max Alekseyev, An explicit representation for polynomials generated by a power of x/sin(x). Answer. MathOverflow 2017. %H A260327 N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 460. %H A260327 N. E. Nörlund, Vorlesungen über Differenzenrechnung, 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE