Real-rooted atlas

A137320 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16\,x^{2} - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(1 + 8\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
226521694
312839233664920
47362888382419202569624
54896227363904030400102401024108336
6371521954404013444086402137605222440961312656
7317376184057643191685300736364672013834242580481638417082432

All 15 rows

Technical details
Node id
n-ec3f66608d5cc6c1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c903033d6bfa763c

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,         4;
  2:       26,        52,        16;
  3:      128,       392,       336,        64;
  4:      736,      2888,      3824,      1920,       256;
  5:     4896,     22736,     39040,     30400,     10240,      1024;
  6:    37152,    195440,    401344,    408640,    213760,     52224,      4096;
  7:   317376,   1840576,   4319168,   5300736,   3646720,   1383424,    258048,    16384;
  8:  3017664,  18942912,  49361856,  69699840,  58145024,  29130752,   8429568,  1245184,   65536;
  9: 31628160, 212118912, 601966464, 949402880, 916210176, 559211520, 214351872, 49053696, 5898240, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+8*n*x+16*x^2)*P(n-2,x) + (2*n+1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 8*n*x + 16*x^2)*P[n - 2, x] + (2*n + 1 + 8*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] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 8*n*x + 16*x^2)*v[i - 2] + (2*n + 1 + 8*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.