Real-rooted atlas

A207606 → x ↦ x+1 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 26 + 15\,x + 2\,x^{2}\)
\(P_{1}(x) = 97 + 84\,x + 23\,x^{2} + 2\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02615243
19784232206
2362418174312987
31351195010912963924729
4504287336140224445047222658
5188173802432202148204005636552108561
67022616218016069289238303906502854632520147
726208768092277274650282420679355762986311047122492174
897812228235953611214269480412996064233399436014216138679211940723
936504011159158016495705138876067686435293720079091615012019689170087257211441
1013623482471931267396318669351325433337401901189660065041377180227490264102046952274116482
11508435271908044063265501653374808802349998501164441244224699611365104226745133143034507242410321313370969
121897506267668680251423041880160712236012341465406817644502794256048633023220207418356676146741244108283411126292738363
13708158977306641832061324853807514050440630870867038447602161757219870613381428164892453341430325401902641420553413276119230150320846
142642885282122069642322617331796034581564780315147385802100598508410594214092412441535012527438222978978205528322879234748599506833437501272144458865867
Technical details
Node id
n-e27ca3f60c180046 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-34678cae11e60551

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      26,       15,        2;
  1:      97,       84,       23,        2;
  2:     362,      418,      174,       31,       2;
  3:    1351,     1950,     1091,      296,      39,       2;
  4:    5042,     8733,     6140,     2244,     450,      47,      2;
  5:   18817,    38024,    32202,    14820,    4005,     636,     55,      2;
  6:   70226,   162180,   160692,    89238,   30390,    6502,    854,     63,     2;
  7:  262087,   680922,   772746,   502824,  206793,   55762,   9863,   1104,    71,    2;
  8:  978122,  2823595,  3611214,  2694804, 1299606,  423339,  94360,  14216,  1386,   79,  2;
  9: 3650401, 11591580, 16495705, 13887606, 7686435, 2937200, 790916, 150120, 19689, 1700, 87, 2;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+x)*P(n-1,x) for n >= 2, with P(0,x) = 26 + 15*x + 2*x^2, P(1,x) = 97 + 84*x + 23*x^2 + 2*x^3.
%t Mathematica
P[0, x_] := 26 + 15*x + 2*x^2;
P[1, x_] := 97 + 84*x + 23*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 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] = 26 + 15*x + 2*x^2; v[2] = 97 + 84*x + 23*x^2 + 2*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 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

Candidate equalities ?

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