Real-rooted atlas

A111125 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-9 + 6\,n\right)\,P_{n}(x) = \left(-27 - 42\,x - 10\,x^{2} + 18\,n + 27\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 18\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
39211141
427909329239
581351540345761393
624312962646256511701998119
77294617117451528210665376252147321
821871603848843797857519540716116791364275807
96561546751939143812674531953324511465203536735711607521
1019683183708743580170950524545432277153136171850494510513593499369319
115904961017327753037304580122982301381382110463418526753816837203080852447654608393
12177147200766610136745300559415806485076626648703771744491571519492056547047089254864079318281039
13531441655443936374184119967885261587070396769914428713236332064846182759895696200401740533425617961677611855077841
141594323212576401286087224670185411134528120194501501124183846002205707598147409704971222409024158406654430920729612343920410812186007
154782969685558894490676451780327350476826579991186615081282861176313475506680106107007056226898571267866457381842491816777579520644185114985163018038201
Technical details
Node id
n-f85b64126b6a6cdc · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-90d3cb4b1288c73a

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,      4;
   3:     9,     21,     11;
   4:    27,     90,     93,      29;
   5:    81,    351,    540,     345,      76;
   6:   243,   1296,   2646,    2565,    1170,     199;
   7:   729,   4617,  11745,   15282,   10665,    3762,     521;
   8:  2187,  16038,  48843,   79785,   75195,   40716,   11679,   1364;
   9:  6561,  54675, 193914,  381267,  453195,  332451,  146520,  35367,   3571;
  10: 19683, 183708, 743580, 1709505, 2454543, 2277153, 1361718, 504945, 105135, 9349;
%F Formula
The row polynomials satisfy (6*n-9)*P(n,x) = (18*n-27+(27*n-42)*x+(5*n-10)*x^2)*P(n-1,x) - (9*x+18*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18*n - 27 + (27*n - 42)*x + (5*n - 10)*x^2)*P[n - 1, x] - (9*x + 18*x^2 + 5*x^3)*D[P[n - 1, x], x])/(6*n - 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((18*n - 27 + (27*n - 42)*x + (5*n - 10)*x^2)*v[i - 1] - (9*x + 18*x^2 + 5*x^3)*deriv(v[i - 1], x))/(6*n - 9)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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