Real-rooted atlas

A207824 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-384\,n - 108\,n x + 54\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-1152 - 1188\,x - 324\,x^{2} - 1152\,n - 1476\,n x - 324\,n x^{2} + 81\,n x^{3} - 288\,n^{2} x + 162\,n^{2} x^{2} + 162\,n^{2} x^{3} + 162\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1024 - 1728\,x - 936\,x^{2} - 162\,x^{3} + 288\,n x^{2} + 324\,n x^{3} + 81\,n x^{4} + 288\,n^{2} x^{2} + 324\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
2357227134
32046364861081434
4118949685751259240514905
56930363065637638448121501458151668
6403912544484958824536002166755248851031518587
7235416173272840639324666680298890010935002143261749615012978

All 15 rows

Technical details
Node id
n-52f453bfa743f0d8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-64ed7e4ec7b0065c

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:       6,        6;
  2:      35,       72,        27;
  3:     204,      636,       486,       108;
  4:    1189,     4968,      5751,      2592,       405;
  5:    6930,    36306,     56376,     38448,     12150,      1458;
  6:   40391,   254448,    495882,    453600,    216675,     52488,     5103;
  7:  235416,  1732728,   4063932,   4666680,   2988900,   1093500,   214326,    17496;
  8: 1372105, 11554416,  31684986,  43802208,  35216775,  17268552,  5108103,   839808,   59049;
  9: 7997214, 75826398, 238040856, 384886512, 372570030, 229298202, 90874224, 22534848, 3188646, 196830;
%F Formula
The row polynomials satisfy -(384*n+108*n*x-54*n^3*x^2)*P(n,x) = -(1152*n+1152+(288*n^2+1476*n+1188)*x-(162*n^3+162*n^2-324*n-324)*x^2-(81*n^3+162*n^2+81*n)*x^3)*P(n-1,x) - (1024+1728*x-(288*n^2+288*n-936)*x^2-(324*n^2+324*n-162)*x^3-(81*n^2+81*n)*x^4)*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[(-(1152*n + 1152 + (288*n^2 + 1476*n + 1188)*x - (162*n^3 + 162*n^2 - 324*n - 324)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*P[n - 1, x] - (1024 + 1728*x - (288*n^2 + 288*n - 936)*x^2 - (324*n^2 + 324*n - 162)*x^3 - (81*n^2 + 81*n)*x^4)*D[P[n - 1, x], x])/(-384*n - 108*n*x + 54*n^3*x^2)]];
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] = (-(1152*n + 1152 + (288*n^2 + 1476*n + 1188)*x - (162*n^3 + 162*n^2 - 324*n - 324)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*v[i - 1] - (1024 + 1728*x - (288*n^2 + 288*n - 936)*x^2 - (324*n^2 + 324*n - 162)*x^3 - (81*n^2 + 81*n)*x^4)*deriv(v[i - 1], x))/(-384*n - 108*n*x + 54*n^3*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