Real-rooted atlas

A207823 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(392\,n + 70\,n x + 50\,n x^{2} - 420\,n^{3} x^{2} - 150\,n^{3} x^{3} + 18\,n^{5} x^{4}\right)\,P_{n}(x) = \left(980 + 1498\,x + 650\,x^{2} - 78\,x^{3} - 90\,x^{4} + 980\,n + 2527\,n x + 545\,n x^{2} - 1149\,n x^{3} - 495\,n x^{4} + 1029\,n^{2} x - 1155\,n^{2} x^{2} - 2391\,n^{2} x^{3} - 810\,n^{2} x^{4} + 27\,n^{2} x^{5} - 1050\,n^{3} x^{2} - 1446\,n^{3} x^{3} - 405\,n^{3} x^{4} + 81\,n^{3} x^{5} - 126\,n^{4} x^{3} + 45\,n^{4} x^{4} + 81\,n^{4} x^{5} + 45\,n^{5} x^{4} + 27\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(343 + 735\,x + 378\,x^{2} - 170\,x^{3} - 201\,x^{4} - 45\,x^{5} - 1029\,n x^{2} - 1995\,n x^{3} - 1191\,n x^{4} - 225\,n x^{5} - 1029\,n^{2} x^{2} - 1995\,n^{2} x^{3} - 1128\,n^{2} x^{4} - 135\,n^{2} x^{5} + 27\,n^{2} x^{6} + 126\,n^{3} x^{4} + 180\,n^{3} x^{5} + 54\,n^{3} x^{6} + 63\,n^{4} x^{4} + 90\,n^{4} x^{5} + 27\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151217
22412081225
311587612154322638
45515640119078640202528763
526403393696390106272506258748298611
6126491957206991111036800749250262440357212991691
76060510964524720275880632085556254540212125023513996829169692
8290376601380030303315681696008330850059398920247546535598720531441278369325
91391275324570601873894504936593607275318756523471083650686201248514562391484519683002610579349
10666599917296680011257290903399537600586837912563452743204464339138204913152059308815698415000714492924130671677
11319387209123689286608781900225079171204454969625056425422312478664972102761372684810721822175268672950039297109525509168220343381226
1215302760147721424803807667067414438688480032238596362546823652504046527195468632357802240015786189184553045685000117105386311530550080898135291991099670391
13733199285247846746842159865332109025020838082244193644375367746456574841904590868003413382819840200717294085084721536900025071555861049436767584583787938531248730817830197610487
1435129688241279496223601209152549493552005171136015129060776100276140028474003550333712278533163094131200228381907064851161700501500043172590438891140769992960203158202598218741115601076168025158409494970039
Technical details
Node id
n-b94ba41f2759b53f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7a6113a2dac57b52

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:       5,       12;
  2:      24,      120,        81;
  3:     115,      876,      1215,       432;
  4:     551,     5640,     11907,      8640,      2025;
  5:    2640,    33936,     96390,    106272,     50625,      8748;
  6:   12649,   195720,    699111,   1036800,    749250,    262440,     35721;
  7:   60605,  1096452,   4720275,   8806320,   8555625,   4540212,   1250235,    139968;
  8:  290376,  6013800,  30303315,  68169600,  83308500,  59398920,  24754653,   5598720,   531441;
  9: 1391275, 32457060, 187389450, 493659360, 727531875, 652347108, 365068620, 124851456, 23914845, 1968300;
%F Formula
The row polynomials satisfy (392*n+70*n*x-(420*n^3-50*n)*x^2-150*n^3*x^3+18*n^5*x^4)*P(n,x) = (980*n+980+(1029*n^2+2527*n+1498)*x-(1050*n^3+1155*n^2-545*n-650)*x^2-(126*n^4+1446*n^3+2391*n^2+1149*n+78)*x^3+(45*n^5+45*n^4-405*n^3-810*n^2-495*n-90)*x^4+(27*n^5+81*n^4+81*n^3+27*n^2)*x^5)*P(n-1,x) + (343+735*x-(1029*n^2+1029*n-378)*x^2-(1995*n^2+1995*n+170)*x^3+(63*n^4+126*n^3-1128*n^2-1191*n-201)*x^4+(90*n^4+180*n^3-135*n^2-225*n-45)*x^5+(27*n^4+54*n^3+27*n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((980*n + 980 + (1029*n^2 + 2527*n + 1498)*x - (1050*n^3 + 1155*n^2 - 545*n - 650)*x^2 - (126*n^4 + 1446*n^3 + 2391*n^2 + 1149*n + 78)*x^3 + (45*n^5 + 45*n^4 - 405*n^3 - 810*n^2 - 495*n - 90)*x^4 + (27*n^5 + 81*n^4 + 81*n^3 + 27*n^2)*x^5)*P[n - 1, x] + (343 + 735*x - (1029*n^2 + 1029*n - 378)*x^2 - (1995*n^2 + 1995*n + 170)*x^3 + (63*n^4 + 126*n^3 - 1128*n^2 - 1191*n - 201)*x^4 + (90*n^4 + 180*n^3 - 135*n^2 - 225*n - 45)*x^5 + (27*n^4 + 54*n^3 + 27*n^2)*x^6)*D[P[n - 1, x], x])/(392*n + 70*n*x - (420*n^3 - 50*n)*x^2 - 150*n^3*x^3 + 18*n^5*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((980*n + 980 + (1029*n^2 + 2527*n + 1498)*x - (1050*n^3 + 1155*n^2 - 545*n - 650)*x^2 - (126*n^4 + 1446*n^3 + 2391*n^2 + 1149*n + 78)*x^3 + (45*n^5 + 45*n^4 - 405*n^3 - 810*n^2 - 495*n - 90)*x^4 + (27*n^5 + 81*n^4 + 81*n^3 + 27*n^2)*x^5)*v[i - 1] + (343 + 735*x - (1029*n^2 + 1029*n - 378)*x^2 - (1995*n^2 + 1995*n + 170)*x^3 + (63*n^4 + 126*n^3 - 1128*n^2 - 1191*n - 201)*x^4 + (90*n^4 + 180*n^3 - 135*n^2 - 225*n - 45)*x^5 + (27*n^4 + 54*n^3 + 27*n^2)*x^6)*deriv(v[i - 1], x))/(392*n + 70*n*x - (420*n^3 - 50*n)*x^2 - 150*n^3*x^3 + 18*n^5*x^4)); 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