Real-rooted atlas

A085880 → basis A013611 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-720 + 480\,x + 768\,n - 512\,n x - 192\,n^{2} + 128\,n^{2} x\right)\,P_{n-2}(x) + \left(48 + 24\,x - 80\,n - 40\,n x + 32\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
37212832232
41080328020803206760
5181447884887808322563584220640
6326592185203230643201935360494592430087715904
76158592431185929634099290157056387932167569408540672282678528
81200925441000826112284114688036096952322278189056736260096115974144702873610709212800

All 15 rows

Technical details
Node id
n-ec43814af8c33829 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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:           6,            4;
   3:          72,          128,            32;
   4:        1080,         3280,          2080,           320;
   5:       18144,        78848,         87808,         32256,          3584;
   6:      326592,      1852032,       3064320,       1935360,        494592,         43008;
   7:     6158592,     43118592,      96340992,      90157056,      38793216,       7569408,        540672;
   8:   120092544,   1000826112,    2841146880,    3609695232,    2278189056,     736260096,     115974144,      7028736;
   9:  2401850880,  23218257920,   80320880640,  130769633280,  111651471360,   52387512320,   13495173120,   1780613120,    93716480;
  10: 48997757952, 538977826816, 2204841852928, 4415976767488, 4832832356352, 3049824780288, 1130520641536, 241526112256, 27402698752, 1274544128;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(192*n^2-768*n+720-(128*n^2-512*n+480)*x)*P(n-2,x) + (32*n^2-80*n+48+(16*n^2-40*n+24)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(192*n^2 - 768*n + 720 - (128*n^2 - 512*n + 480)*x)*P[n - 2, x] + (32*n^2 - 80*n + 48 + (16*n^2 - 40*n + 24)*x)*P[n - 1, x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(192*n^2 - 768*n + 720 - (128*n^2 - 512*n + 480)*x)*v[i - 2] + (32*n^2 - 80*n + 48 + (16*n^2 - 40*n + 24)*x)*v[i - 1])/(n^2 - n)); 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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

Same fingerprint ?

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.