Real-rooted atlas

A094816 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}(x) + \left(4 + n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31620541
4641449216316
52568961040464652721
61024512092807520267632625946
7409627648716809056058144177161957271801
816384143360501760904960901376490896133500137003105936
965536720896326860879421441124838494420484545056113228810960138474561
10262144353894420152320633077761204008961430123521053896964605254410693284986410513796366
11104857617039360118947840468418560114857164818249799681893459200125851840050844561611138117298641017360674441
124194304807403526776422403265658880100137861122044970188828269587456263254604801608210630460863466401269097628108505112112632827396
131677721637748736037486592002168455168081200283648206825422848366290186240452731699200385731360256219569066112786059950561570709672013020613451833790646881
146710886417448304642022912819213825055129662025315123219249177559044240551231488669359298150475419341803525962609782272319540563014410902761579842098784279241692679748631656637715106
15268435456798595481610687086592085153179238445060383047681672156912025644780367577088876939099013121258098590228481310939397447689720626157132849437791224640161717781457283011691351444236975164805577636838177561
Technical details
Node id
n-c5e11caa6c1bab83 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1a85bd647d097269

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:      4,       2;
   3:     16,      20,        5;
   4:     64,     144,       92,       16;
   5:    256,     896,     1040,      464,        65;
   6:   1024,    5120,     9280,     7520,      2676,       326;
   7:   4096,   27648,    71680,    90560,     58144,     17716,      1957;
   8:  16384,  143360,   501760,   904960,    901376,    490896,    133500,    13700;
   9:  65536,  720896,  3268608,  7942144,  11248384,   9442048,   4545056,  1132288,   109601;
  10: 262144, 3538944, 20152320, 63307776, 120400896, 143012352, 105389696, 46052544, 10693284, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*x^2*P(n-2,x) + (4+n*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*x^2*P[n - 2, x] + (4 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*x^2*v[i - 2] + (4 + n*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, 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