A260326 -id:A260326 - cp's OEIS frontend

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

N. J. A. Sloane, Jul 25 2015

			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;

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]

For denominators see A260326.

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

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 *)

{ 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

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

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

More terms by Peter Luschny, Jul 01 2019

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

Original entry on oeis.org

3, 5, -6, 16, 1, 7, 16, 28, 0, 15, 225, 1265, 3707, 7120, 4900, -6480, 27648, 3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0, 63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400, 9, 531, 14010, 219106

Offset: 1

Gregory Gerard Wojnar, Aug 09 2017

The polynomials come in pairs: first of odd degree; second of even degree 1 greater, whose constant term is always zero. Observations: All coefficients are positive except for the linear coefficients of the first polynomial in each pair, which are always negative. From the first of one pair to the first of the next pair, the degree always grows by 4. The "standard" factors of polynomials yielding the columns of triangle A290053 (beginning with column 3) are always of the form (1/A053657(k+2))*(N + k + 2) in odd rows of this triangle A290761, and of the form (N/A053657(k+2))*(N + k + 3)^2 in even rows of this triangle, where k is the row number. See examples.

			The first rows of the triangle are parsed as follows:
3, 5, -6, 16;
1, 7, 16, 28, 0;
15, 225, 1265, 3707, 7120, 4900, -6480, 27648;
3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0;
63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400;
9, 531, 14010, 219106, 2266137, 16325259, 83797380, 307998768, 802828704, 1433652560, 1651979520, 1239918336, 0.
The associated full polynomials giving the columns of triangle A290053 are then:
(1/24) * (N + 3) * (3*N^3 + 5*N^2 - 6*N + 16);
(N/48) * (N + 5)^2 * (1*N^3 + 7*N^2 + 16*N + 28);
(1/5760) * (N + 5) * (15*N^7 + 225*N^6 + 1265*N^5 + 3707*N^4 + 7120*N^3 + 4900*N^2 - 6480*N + 27648);
(N/11520) * (N + 7)^2 * (3*N^7 + 83*N^6 + 961*N^5 + 6201*N^4 + 24708*N^3 + 60700*N^2 + 87968*N + 85056); etc.

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

The first column of this triangle is A290030; alternating entries of the first column give A260326. See also triangle A290053, whose columns are A000012-A000096, A290061-A290071, A290127-A290723, etc.

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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).

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