Real-rooted atlas

A111125 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-600 - 480\,x + 2496\,x^{2} + 1920\,x^{3} - 384\,x^{4} + 400\,n + 320\,n x - 5264\,n x^{2} - 3872\,n x^{3} + 1408\,n x^{4} + 3600\,n^{2} x^{2} + 2592\,n^{2} x^{3} - 2016\,n^{2} x^{4} - 800\,n^{3} x^{2} - 576\,n^{3} x^{3} + 1408\,n^{3} x^{4} - 480\,n^{4} x^{4} + 64\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-400 - 520\,x + 384\,x^{2} + 736\,x^{3} + 256\,x^{4} + 600\,n - 20\,n x - 3616\,n x^{2} - 4464\,n x^{3} - 1408\,n x^{4} + 700\,n^{2} x + 4400\,n^{2} x^{2} + 5088\,n^{2} x^{3} + 1088\,n^{2} x^{4} - 256\,n^{2} x^{5} - 1200\,n^{3} x^{2} - 1264\,n^{3} x^{3} + 608\,n^{3} x^{4} + 512\,n^{3} x^{5} - 160\,n^{4} x^{3} - 640\,n^{4} x^{4} - 320\,n^{4} x^{5} + 96\,n^{5} x^{4} + 64\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 600\,x + 640\,x^{2} - 512\,x^{3} - 912\,x^{4} - 256\,x^{5} + 1400\,n x^{2} + 4800\,n x^{3} + 4576\,n x^{4} + 1152\,n x^{5} - 700\,n^{2} x^{2} - 2400\,n^{2} x^{3} - 1968\,n^{2} x^{4} + 192\,n^{2} x^{5} + 256\,n^{2} x^{6} - 320\,n^{3} x^{4} - 768\,n^{3} x^{5} - 256\,n^{3} x^{6} + 80\,n^{4} x^{4} + 192\,n^{4} x^{5} + 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
24812
3115636103
429248360128765
576920216016644005220
6199312010260125446400115233675
752110032426607244858000218883136208685
81364311441628643564803940002315526899281921254588

All 15 rows

Technical details
Node id
n-8084105d325fd2bf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2a39904a34ff3934

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,      8;
   3:   11,     56,      36;
   4:   29,    248,     360,     128;
   5:   76,    920,    2160,    1664,      400;
   6:  199,   3120,   10260,   12544,     6400,     1152;
   7:  521,  10032,   42660,   72448,    58000,    21888,    3136;
   8: 1364,  31144,  162864,  356480,   394000,   231552,   68992,    8192;
   9: 3571,  94312,  586080, 1576064,  2238000,  1807488,  834176,  204800,  20736;
  10: 9349, 280360, 2019780, 6455552, 11245200, 11636352, 7353920, 2785280, 580608, 51200;
%F Formula
The row polynomials satisfy (400*n-600+(320*n-480)*x-(800*n^3-3600*n^2+5264*n-2496)*x^2-(576*n^3-2592*n^2+3872*n-1920)*x^3+(64*n^5-480*n^4+1408*n^3-2016*n^2+1408*n-384)*x^4)*P(n,x) = (600*n-400+(700*n^2-20*n-520)*x-(1200*n^3-4400*n^2+3616*n-384)*x^2-(160*n^4+1264*n^3-5088*n^2+4464*n-736)*x^3+(96*n^5-640*n^4+608*n^3+1088*n^2-1408*n+256)*x^4+(64*n^5-320*n^4+512*n^3-256*n^2)*x^5)*P(n-1,x) + (125+600*x-(700*n^2-1400*n-640)*x^2-(2400*n^2-4800*n+512)*x^3+(80*n^4-320*n^3-1968*n^2+4576*n-912)*x^4+(192*n^4-768*n^3+192*n^2+1152*n-256)*x^5+(64*n^4-256*n^3+256*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((600*n - 400 + (700*n^2 - 20*n - 520)*x - (1200*n^3 - 4400*n^2 + 3616*n - 384)*x^2 - (160*n^4 + 1264*n^3 - 5088*n^2 + 4464*n - 736)*x^3 + (96*n^5 - 640*n^4 + 608*n^3 + 1088*n^2 - 1408*n + 256)*x^4 + (64*n^5 - 320*n^4 + 512*n^3 - 256*n^2)*x^5)*P[n - 1, x] + (125 + 600*x - (700*n^2 - 1400*n - 640)*x^2 - (2400*n^2 - 4800*n + 512)*x^3 + (80*n^4 - 320*n^3 - 1968*n^2 + 4576*n - 912)*x^4 + (192*n^4 - 768*n^3 + 192*n^2 + 1152*n - 256)*x^5 + (64*n^4 - 256*n^3 + 256*n^2)*x^6)*D[P[n - 1, x], x])/(400*n - 600 + (320*n - 480)*x - (800*n^3 - 3600*n^2 + 5264*n - 2496)*x^2 - (576*n^3 - 2592*n^2 + 3872*n - 1920)*x^3 + (64*n^5 - 480*n^4 + 1408*n^3 - 2016*n^2 + 1408*n - 384)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((600*n - 400 + (700*n^2 - 20*n - 520)*x - (1200*n^3 - 4400*n^2 + 3616*n - 384)*x^2 - (160*n^4 + 1264*n^3 - 5088*n^2 + 4464*n - 736)*x^3 + (96*n^5 - 640*n^4 + 608*n^3 + 1088*n^2 - 1408*n + 256)*x^4 + (64*n^5 - 320*n^4 + 512*n^3 - 256*n^2)*x^5)*v[i - 1] + (125 + 600*x - (700*n^2 - 1400*n - 640)*x^2 - (2400*n^2 - 4800*n + 512)*x^3 + (80*n^4 - 320*n^3 - 1968*n^2 + 4576*n - 912)*x^4 + (192*n^4 - 768*n^3 + 192*n^2 + 1152*n - 256)*x^5 + (64*n^4 - 256*n^3 + 256*n^2)*x^6)*deriv(v[i - 1], x))/(400*n - 600 + (320*n - 480)*x - (800*n^3 - 3600*n^2 + 5264*n - 2496)*x^2 - (576*n^3 - 2592*n^2 + 3872*n - 1920)*x^3 + (64*n^5 - 480*n^4 + 1408*n^3 - 2016*n^2 + 1408*n - 384)*x^4)); 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. [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