Real-rooted atlas

A178822 → 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

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-200 + 60\,x - 90\,n + 27\,n x - 10\,n^{2} + 3\,n^{2} x\right)\,P_{n-2}(x) + \left(-35 - 15\,x + 28\,n + 12\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 30 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1301848
25258191891533
370001864810584151237744
4787503012662959749865810206784854
57875003945564572670032319007552446123614508144
67218750447858188765479872611154281787665051970336798245838054
761875000457976376113940367212745505527183099442107393203060050417321043895187472

All 15 rows

Technical details
Node id
n-75d573c0ca4216ce · 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:         30,          18;
  2:        525,         819,          189;
  3:       7000,       18648,        10584,         1512;
  4:      78750,      301266,       295974,        98658,        10206;
  5:     787500,     3945564,      5726700,      3231900,       755244,        61236;
  6:    7218750,    44785818,     87654798,     72611154,     28178766,      5051970,      336798;
  7:   61875000,   457976376,   1139403672,   1274550552,    718309944,    210739320,    30600504,    1732104;
  8:  502734375,  4323845097,  13125820851,  18762409809,  14206051431,   5993681265,  1408272723,  171694809,   8444007;
  9: 3910156250, 38320556274, 137774943306, 241978298562, 233754090570, 131679598050, 44129631546, 8629775154, 906323418, 39405366;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(10*n^2+90*n+200-(3*n^2+27*n+60)*x)*P(n-2,x) + (7*n^2+28*n-35+(3*n^2+12*n-15)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 30 + 18*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 30 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(10*n^2 + 90*n + 200 - (3*n^2 + 27*n + 60)*x)*P[n - 2, x] + (7*n^2 + 28*n - 35 + (3*n^2 + 12*n - 15)*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] = 30 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = (-(10*n^2 + 90*n + 200 - (3*n^2 + 27*n + 60)*x)*v[i - 2] + (7*n^2 + 28*n - 35 + (3*n^2 + 12*n - 15)*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.