Real-rooted atlas

A137320 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2} - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21618438
376124608268
4428876620176162116
528086728604025204803218608
6210965688860376322009040124864180912
71788485294086376164049921461602979231361281930080

All 15 rows

Technical details
Node id
n-bc6dfc9c53b05296 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9d29ecf489bb5c8f

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:        4,        2;
  2:       16,       18,        4;
  3:       76,      124,       60,        8;
  4:      428,      876,      620,      176,       16;
  5:     2808,     6728,     6040,     2520,      480,       32;
  6:    21096,    56888,    60376,    32200,     9040,     1248,      64;
  7:   178848,   529408,   637616,   404992,   146160,    29792,    3136,    128;
  8:  1690272,  5399424,  7188240,  5214944,  2254224,   593152,   92288,   7680,   256;
  9: 17628480, 60029568, 86753280, 69960736, 34720896, 10991904, 2214912, 272640, 18432, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+4*n*x+4*x^2)*P(n-2,x) + (2*n+1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n*x + 4*x^2)*P[n - 2, x] + (2*n + 1 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n*x + 4*x^2)*v[i - 2] + (2*n + 1 + 4*x)*v[i - 1]); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.