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

0,5

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]

Formula

E.g.f. Sum_{n>=0} D_{2n}(x) y^(2n)/(2n)! = (y/sinh(y))^x. - Max Alekseyev, Jul 04 2019

Example

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;

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 *)

Prog

(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

Crossrefs

For denominators see A260326.

Sequence in context: A088307 A369629 A324611 * A329251 A062627 A011217

Adjacent sequences: A260324 A260325 A260326 * A260328 A260329 A260330

Keyword

sign,tabl,look,frac

Author

N. J. A. Sloane, Jul 25 2015

Extensions

Typo in data and example corrected by Jean-François Alcover, Jul 01 2019

More terms by Peter Luschny, Jul 01 2019

Status

approved