Real-rooted atlas

A207824 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(9 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
191221
280216144440
37112892388817289219
4631934344695526220820736193159
556160382032103420813927689331202488324047120
6499121407786413822704248832002509056013436928298598484796361
7443592942308196172304496388428480523480320421770240188116992358318081776676461

All 15 rows

Technical details
Node id
n-4057433cde1b04ed · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-15f20b2c09fe3d56

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:         9,         12;
  2:        80,        216,         144;
  3:       711,       2892,        3888,        1728;
  4:      6319,      34344,       69552,       62208,        20736;
  5:     56160,     382032,     1034208,     1392768,       933120,       248832;
  6:    499121,    4077864,    13822704,    24883200,     25090560,     13436928,      2985984;
  7:   4435929,   42308196,   172304496,   388428480,    523480320,    421770240,    188116992,     35831808;
  8:  39424240,  429927048,  2044616112,  5538627072,   9347374080,  10064259072,   6751309824,   2579890176,   429981696;
  9: 350382231, 4300126116, 23388365088, 73994608512, 150066411264, 202325050368, 181344780288, 104198897664, 34828517376, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (9+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 12*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (9 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (9 + 12*x)*v[i - 1]); 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023