Real-rooted atlas

A063007 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 3\,n\right)\,P_{n}(x) = \left(9\,n + 18\,n x\right)\,P_{n-1}(x) + \left(4 + 18\,x + 18\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
266
354108162
4396162016203636
5270016740340202268076140
6177301474204536006123603061801537290
7113778118692048988801000188010103940404157630346974
871887290221044674348012798324019534284015762146452540488589972488

All 15 rows

Technical details
Node id
n-8fb26d3bf722c8bb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6a81dd80276de4b8

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 -2)
  • 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:        0;
   2:        6;
   3:       54,       108;
   4:      396,      1620,       1620;
   5:     2700,     16740,      34020,       22680;
   6:    17730,    147420,     453600,      612360,      306180;
   7:   113778,   1186920,    4898880,    10001880,    10103940,     4041576;
   8:   718872,   9022104,   46743480,   127983240,   195342840,   157621464,    52540488;
   9:  4491288,  65885400,  410893560,  1412306280,  2889726840,  3520212696,  2364321960,   675520560;
  10: 27823230, 467133480, 3406898880, 14098364280, 36209152980, 59108049000, 59896156320, 34451548560, 8612887140;
%F Formula
The row polynomials satisfy (3*n-6)*P(n,x) = (9*n+18*n*x)*P(n-1,x) + (4+18*x+18*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + 18*n*x)*P[n - 1, x] + (4 + 18*x + 18*x^2)*D[P[n - 1, x], x])/(3*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = ((9*n + 18*n*x)*v[i - 1] + (4 + 18*x + 18*x^2)*deriv(v[i - 1], x))/(3*n - 6)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023