Real-rooted atlas

A116414 → basis A013611 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-8 + 8\,n\right)\,P_{n}(x) = \left(40 + 32\,x + 20\,n + 16\,n x\right)\,P_{n-1}(x) + \left(13 + 40\,x + 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2403272
3276480192948
4164045123840102411016
590083424045696256005120119664
647160229728422400368640153600245761246104
7239068142320033532804019200259584086016011468812605436
811842008339360240384003687424032614400166871044587520524288124849512
95764620469008001601683203005184003396556802374041601004666882359296023592961216830924
1027683600255602880101013120022462003203103027200276730675215964569605756682241179648001048576011710527696
11131507168135878080061037683201571681280025727170560280188518402059498291210105651200317325312057671680046137344111553632864
12619160880707915270435641132800104398894080197910835200255282757632228940185600142949744640609878016001695547392027682406402013265921053734706288
13289351301236272708160202369275264664930073600143436713472021423292416002274235514880173101547520093869309952035410411520088306876416130862284808724152329883475671284
1413437205480183274186784112255922688040907564871689902753536000168307601817602068189347840018622733352960123000717312005891623485440199128776704045016625971261069066240375809638492146144061448
Technical details
Node id
n-d688f3b94a5053d6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a406a0b215f7429e

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       4;
  2:      40,       32;
  3:     276,      480,       192;
  4:    1640,     4512,      3840,      1024;
  5:    9008,    34240,     45696,     25600,      5120;
  6:   47160,   229728,    422400,    368640,    153600,     24576;
  7:  239068,  1423200,   3353280,   4019200,   2595840,    860160,    114688;
  8: 1184200,  8339360,  24038400,  36874240,  32614400,  16687104,   4587520,   524288;
  9: 5764620, 46900800, 160168320, 300518400, 339655680, 237404160, 100466688, 23592960, 2359296;
%F Formula
The row polynomials satisfy (8*n-8)*P(n,x) = (20*n+40+(16*n+32)*x)*P(n-1,x) + (13+40*x+16*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (16*n + 32)*x)*P[n - 1, x] + (13 + 40*x + 16*x^2)*D[P[n - 1, x], x])/(8*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 40 + (16*n + 32)*x)*v[i - 1] + (13 + 40*x + 16*x^2)*deriv(v[i - 1], x))/(8*n - 8)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023