Real-rooted atlas

A133214 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 2\,x + 4\,x^{2} + 11\,n + 9\,n x + 2\,n x^{2} + 8\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(18 + 18\,x + 28\,x^{2} + 16\,x^{3} + 25\,n + 65\,n x + 52\,n x^{2} + 12\,n x^{3} + 12\,n^{2} + 52\,n^{2} x + 24\,n^{2} x^{2} + 2\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 14\,x^{2} - 16\,x^{3} - 8\,x^{4} + 14\,n x - 20\,n x^{2} - 16\,n x^{3} + 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 9 + 16\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0916429
11990608177
233320528192161089
35185028002400560326693
47318721050017600936015366441005
5993626311648820090160329284032128250337
6129640078400338688580160401408107520102402561523201
7163105301745281072512279417632175361612800331776253445129239877
8201164003537002937600108662401896652815724800599040097920061440102455897533
92432444266550071874003589344088792704111234816696960002090880027878401464322048337339665
102893513611790241606880010428264034792243261667020858210099228575360069386240770457634406440962031452097
11339490101987440333727682731986401183751712282686976037691596802779755264109933824022081945620766720798720819212209875941
123936664032130286518803265731265635911715841113079968019976546304207774927361231258828840109189126783795205479219218350081638473260321357
1345188650501270012090260014726930409907785888387058672009010647360012603127680010517002496051209174016139915776002022092800141926400417792032768438889107393
1451311571275840002145024003105706240252357181441213669416963559761920006454685952007236786585604954884014082018857451524696473600058720256003617587209437184655362625636184065
Technical details
Node id
n-af0fa7591e93aaa0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e8b03ce9b132c5b4

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:   9,    16,      4;
  1:  19,    90,     60,       8;
  2:  33,   320,    528,     192,       16;
  3:  51,   850,   2800,    2400,      560,       32;
  4:  73,  1872,  10500,   17600,     9360,     1536,        64;
  5:  99,  3626,  31164,   88200,    90160,    32928,      4032,      128;
  6: 129,  6400,  78400,  338688,   580160,   401408,    107520,    10240,      256;
  7: 163, 10530, 174528, 1072512,  2794176,  3217536,   1612800,   331776,    25344,     512;
  8: 201, 16400, 353700, 2937600, 10866240, 18966528,  15724800,  5990400,   979200,   61440,   1024;
  9: 243, 24442, 665500, 7187400, 35893440, 88792704, 111234816, 69696000, 20908800, 2787840, 146432, 2048;
%F Formula
The row polynomials satisfy (2*n^3+8*n^2+11*n+6+(4*n^2+9*n+2)*x+(2*n+4)*x^2)*P(n,x) = (2*n^3+12*n^2+25*n+18+(12*n^3+52*n^2+65*n+18)*x+(24*n^2+52*n+28)*x^2+(12*n+16)*x^3)*P(n-1,x) + ((4*n^2+14*n+12)*x-(8*n^2+20*n+14)*x^2-(16*n+16)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 9 + 16*x + 4*x^2.
%t Mathematica
P[0, x_] := 9 + 16*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 12*n^2 + 25*n + 18 + (12*n^3 + 52*n^2 + 65*n + 18)*x + (24*n^2 + 52*n + 28)*x^2 + (12*n + 16)*x^3)*P[n - 1, x] + ((4*n^2 + 14*n + 12)*x - (8*n^2 + 20*n + 14)*x^2 - (16*n + 16)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 8*n^2 + 11*n + 6 + (4*n^2 + 9*n + 2)*x + (2*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 9 + 16*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 12*n^2 + 25*n + 18 + (12*n^3 + 52*n^2 + 65*n + 18)*x + (24*n^2 + 52*n + 28)*x^2 + (12*n + 16)*x^3)*v[i - 1] + ((4*n^2 + 14*n + 12)*x - (8*n^2 + 20*n + 14)*x^2 - (16*n + 16)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 8*n^2 + 11*n + 6 + (4*n^2 + 9*n + 2)*x + (2*n + 4)*x^2)); 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. [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