Real-rooted atlas

A111125 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(120 + 192\,x - 384\,x^{2} - 80\,n - 128\,n x + 832\,n x^{2} - 576\,n^{2} x^{2} + 128\,n^{3} x^{2}\right)\,P_{n}(x) = \left(80 + 288\,x + 256\,x^{2} - 120\,n - 272\,n x + 256\,n x^{2} + 512\,n x^{3} - 80\,n^{2} x - 704\,n^{2} x^{2} - 768\,n^{2} x^{3} + 192\,n^{3} x^{2} + 256\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 200\,x - 464\,x^{2} - 256\,x^{3} - 160\,n x^{2} - 768\,n x^{3} - 512\,n x^{4} + 80\,n^{2} x^{2} + 384\,n^{2} x^{3} + 256\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
24812
3115648115
4292484802561013
5769202880332812808484
61993120136802508820480614468711
7521100325688014489618560011673628672543337
8136431144217152712960126080012349446307841310724220220

All 15 rows

Technical details
Node id
n-4f5b30e5043fd166 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c7d6d1540a0cf547

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,      48;
   4:   29,    248,     480,      256;
   5:   76,    920,    2880,     3328,     1280;
   6:  199,   3120,   13680,    25088,    20480,     6144;
   7:  521,  10032,   56880,   144896,   185600,   116736,    28672;
   8: 1364,  31144,  217152,   712960,  1260800,  1234944,   630784,   131072;
   9: 3571,  94312,  781440,  3152128,  7161600,  9639936,  7626752,  3276800,   589824;
  10: 9349, 280360, 2693040, 12911104, 35984640, 62060544, 67235840, 44564480, 16515072, 2621440;
%F Formula
The row polynomials satisfy -(80*n-120+(128*n-192)*x-(128*n^3-576*n^2+832*n-384)*x^2)*P(n,x) = -(120*n-80+(80*n^2+272*n-288)*x-(192*n^3-704*n^2+256*n+256)*x^2-(256*n^3-768*n^2+512*n)*x^3)*P(n-1,x) - (25+200*x-(80*n^2-160*n-464)*x^2-(384*n^2-768*n-256)*x^3-(256*n^2-512*n)*x^4)*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[(-(120*n - 80 + (80*n^2 + 272*n - 288)*x - (192*n^3 - 704*n^2 + 256*n + 256)*x^2 - (256*n^3 - 768*n^2 + 512*n)*x^3)*P[n - 1, x] - (25 + 200*x - (80*n^2 - 160*n - 464)*x^2 - (384*n^2 - 768*n - 256)*x^3 - (256*n^2 - 512*n)*x^4)*D[P[n - 1, x], x])/(-80*n + 120 - (128*n - 192)*x + (128*n^3 - 576*n^2 + 832*n - 384)*x^2)]];
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] = (-(120*n - 80 + (80*n^2 + 272*n - 288)*x - (192*n^3 - 704*n^2 + 256*n + 256)*x^2 - (256*n^3 - 768*n^2 + 512*n)*x^3)*v[i - 1] - (25 + 200*x - (80*n^2 - 160*n - 464)*x^2 - (384*n^2 - 768*n - 256)*x^3 - (256*n^2 - 512*n)*x^4)*deriv(v[i - 1], x))/(-80*n + 120 - (128*n - 192)*x + (128*n^3 - 576*n^2 + 832*n - 384)*x^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. [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