Real-rooted atlas

A116414 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(-10\,x - 13\,x^{2} + 8\,n + 30\,n x + 13\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 20\,x^{2} - 13\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1257
24202246
386013895301
4161605648204091969
532400190442804504176012880
6649605760176002871623580757384253
71282240162246280013972017790011953432585551131
8256512043456203840576160100160010424205921001402063605158
9512115201121286182402122848469560066736805862600288231060327523582713
101024256002810881781760720652819393920350968804208880031950360138418002595757154263517
112048563206886404934400229854727296576016079481624557520025432368016984760065753584111689601009096480
12409612288016558081323520069799680255513600664798848123694272016312327201485047200884894088309580440480575296600884809
1381922662403919872345804802037094408452224002538209280557898240089647945921038953240084320531364534088520144675650620678076543178904223
1416384573440915865688391680575353856266928640090931315202308247040043830104640618922096006391807011246773301120229092733486718602740889731238282449675134
Technical details
Node id
n-8024621785cc762c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d7a31a2c4d4046d1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:   1;
  1:   2,     5;
  2:   4,    20,     22;
  3:   8,    60,    138,     95;
  4:  16,   160,    564,    820,     409;
  5:  32,   400,   1904,   4280,    4504,    1760;
  6:  64,   960,   5760,  17600,   28716,   23580,    7573;
  7: 128,  2240,  16224,  62800,  139720,  177900,  119534,   32585;
  8: 256,  5120,  43456, 203840,  576160, 1001600, 1042420,  592100,  140206;
  9: 512, 11520, 112128, 618240, 2122848, 4695600, 6673680, 5862600, 2882310, 603275;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (8*n+(30*n-10)*x+(13*n-13)*x^2)*P(n-1,x) - (4*x+20*x^2+13*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + (30*n - 10)*x + (13*n - 13)*x^2)*P[n - 1, x] - (4*x + 20*x^2 + 13*x^3)*D[P[n - 1, x], x])/(4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + (30*n - 10)*x + (13*n - 13)*x^2)*v[i - 1] - (4*x + 20*x^2 + 13*x^3)*deriv(v[i - 1], x))/(4*n)); 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

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023