Real-rooted atlas

A178820 → basis A013610 → basis A008288

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-60 + 18\,x - 50\,n + 15\,n x - 10\,n^{2} + 3\,n^{2} x\right)\,P_{n-2}(x) + \left(-21 - 9\,x + 14\,n + 6\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 20 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1201232
225039090730
325006660378054013480
4218758368582215274052835218015
51750008767921272600718200167832136083224032
613125008142876159372361320202851234129185406123644697828
7937500069390360172636920193113720108834840319302004636440262440590179920

All 15 rows

Technical details
Node id
n-9dd73c3d712d6411 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-0fee5d4c7376c95e

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:        20,         12;
  2:       250,        390,          90;
  3:      2500,       6660,        3780,         540;
  4:     21875,      83685,       82215,       27405,        2835;
  5:    175000,     876792,     1272600,      718200,      167832,       13608;
  6:   1312500,    8142876,    15937236,    13202028,     5123412,      918540,      61236;
  7:   9375000,   69390360,   172636920,   193113720,   108834840,    31930200,    4636440,    262440;
  8:  64453125,  554339115,  1682797545,  2405437155,  1821288645,   768420675,  180547785,  22012155,  1082565;
  9: 429687500, 4211050140, 15140103660, 26591021820, 25687262700, 14470285500, 4849410060, 948326940, 99595980, 4330260;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(10*n^2+50*n+60-(3*n^2+15*n+18)*x)*P(n-2,x) + (7*n^2+14*n-21+(3*n^2+6*n-9)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 20 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 20 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(10*n^2 + 50*n + 60 - (3*n^2 + 15*n + 18)*x)*P[n - 2, x] + (7*n^2 + 14*n - 21 + (3*n^2 + 6*n - 9)*x)*P[n - 1, x])/(n^2 - n)]];
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] = 20 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(10*n^2 + 50*n + 60 - (3*n^2 + 15*n + 18)*x)*v[i - 2] + (7*n^2 + 14*n - 21 + (3*n^2 + 6*n - 9)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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.