Real-rooted atlas

A207824 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-128\,n - 12\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-384 - 132\,x - 12\,x^{2} - 384\,n - 164\,n x - 12\,n x^{2} + n x^{3} - 32\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{2} x^{3} + 6\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1024 - 576\,x - 104\,x^{2} - 6\,x^{3} + 32\,n x^{2} + 12\,n x^{3} + n x^{4} + 32\,n^{2} x^{2} + 12\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
23524362
3204212544474
4118916566399653585
569301210262641424150626876
64039184816550981680026752167200003
723541657757645154817284036900450029481479082

All 15 rows

Technical details
Node id
n-db0b998c981443e4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7e6e5b4f10139d04

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,        2;
  2:      35,       24,        3;
  3:     204,      212,       54,        4;
  4:    1189,     1656,      639,       96,       5;
  5:    6930,    12102,     6264,     1424,     150,      6;
  6:   40391,    84816,    55098,    16800,    2675,    216,      7;
  7:  235416,   577576,   451548,   172840,   36900,   4500,    294,     8;
  8: 1372105,  3851472,  3520554,  1622304,  434775,  71064,   7007,   384,   9;
  9: 7997214, 25275466, 26448984, 14255056, 4599630, 943614, 124656, 10304, 486, 10;
%F Formula
The row polynomials satisfy -(128*n+12*n*x-2*n^3*x^2)*P(n,x) = -(384*n+384+(32*n^2+164*n+132)*x-(6*n^3+6*n^2-12*n-12)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (1024+576*x-(32*n^2+32*n-104)*x^2-(12*n^2+12*n-6)*x^3-(n^2+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[(-(384*n + 384 + (32*n^2 + 164*n + 132)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (1024 + 576*x - (32*n^2 + 32*n - 104)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-128*n - 12*n*x + 2*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] = (-(384*n + 384 + (32*n^2 + 164*n + 132)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (1024 + 576*x - (32*n^2 + 32*n - 104)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-128*n - 12*n*x + 2*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