Real-rooted atlas

A136532 → x ↦ x+1

Proved for all n all roots realinterlacing (weak)

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-6 - 11\,n - 6\,n^{2} - n^{3}\right)\,P_{n-2}(x) + \left(3 - 3\,x + 7\,n - n x + 2\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = -20 - 8\,x + 4\,x^{2}\)
\(P_{2}(x) = -80 + 20\,x + 35\,x^{2} - 5\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1-20-84-24
2-802035-5-30
3-192456174-846360
4784431270-910161-74410
51907231424-12400-74882776-272833120
6229536166464-221112-3823238880-6660423-9169290
722998405440-2878160207520468400-13244013810-62010-16200
820282944-18908032-31937488108315684681688-2332528362054-25894869-11-17044830
9144429312-414691584-29597088023444736029014368-375614408391768-85728045060-117612-332754480
10385305856-6886850048-17170275204105398336-263389152-552403488179830560-248913601829568-739961547-13-4772269710
11-14720394752-1013287367681461026470464149508864-15819558720-70886376963639343680-66618048064552992-3605560115990-198814-57163329720
12-469430929920-1355454981120834733246080902083885440-441108912000-6520601664070010429760-168896016002086097040-1513122006668310-1749902505-15-539321599350
13-10222242357248-158387400949762221343683993610720143785984-101225783234561804377374721270640477184-41187389030463635299584-5775316096327465248-11707264255664-310416-2152599831360
14-195267955503104-13152902648627248842320623104076681262976000-2108093248040963294251149772821107235018368-97340713440001866423931680-20697982201614525877256-66403944819677840-3634603791-1773487162851290
15-34653776817438724510645838069769773569693255680-1087605702807552-40853954413301761252533299973120291989185705728-22323573372672053242318312128-7106730814656601635673584-338047669441275365520-31871448504666-4572182954246865531480
Technical details
Node id
n-eea87339949eb904 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5e6f94e8e9fb2fe6

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 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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:       -20,          -8,           4;
   2:       -80,          20,          35,         -5;
   3:      -192,         456,         174,        -84,          6;
   4:       784,        4312,          70,       -910,        161,         -7;
   5:     19072,       31424,      -12400,      -7488,       2776,       -272,         8;
   6:    229536,      166464,     -221112,     -38232,      38880,      -6660,       423,        -9;
   7:   2299840,        5440,    -2878160,     207520,     468400,    -132440,     13810,      -620,      10;
   8:  20282944,   -18908032,   -31937488,   10831568,    4681688,   -2332528,    362054,    -25894,     869,    -11;
   9: 144429312,  -414691584,  -295970880,  234447360,   29014368,  -37561440,   8391768,   -857280,   45060,  -1176,   12;
  10: 385305856, -6886850048, -1717027520, 4105398336, -263389152, -552403488, 179830560, -24891360, 1829568, -73996, 1547, -13;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(n^3+6*n^2+11*n+6)*P(n-2,x) + (2*n^2+7*n+3-(n+3)*x)*P(n-1,x) for n >= 3, with P(1,x) = -20 - 8*x + 4*x^2, P(2,x) = -80 + 20*x + 35*x^2 - 5*x^3.
%t Mathematica
P[1, x_] := -20 - 8*x + 4*x^2;
P[2, x_] := -80 + 20*x + 35*x^2 - 5*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 6*n^2 + 11*n + 6)*P[n - 2, x] + (2*n^2 + 7*n + 3 - (n + 3)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = -20 - 8*x + 4*x^2; v[2] = -80 + 20*x + 35*x^2 - 5*x^3; for(n = 3, N, my(i = n); v[i] = (-(n^3 + 6*n^2 + 11*n + 6)*v[i - 2] + (2*n^2 + 7*n + 3 - (n + 3)*x)*v[i - 1])/(n + 2)); 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

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