Real-rooted atlas

basis-map coefficients of A030528, shifted → basis A013610 → basis A008288

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

Recurrence

\(P_{n}(x) = \left(-4 + 6\,x\right)\,P_{n-2}(x) + \left(-5 - 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = -4 - 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1-4-3-7
21633958
3-64-225-162-27-478
425612811647675813940
5-1024-6657-12780-9180-2592-243-32476
64096327698496991422442269477729267688
7-16384-155649-511974-751977-540108-194643-33534-2187-2206456
8655367208972883555543993353280992820987805545115911656118187024
9-262144-3276801-15466464-35914266-45311724-32551308-13524408-3188646-393660-19683-149909104
101048576146800657995388522151232034562862031937635817897970660886080122034601318761590491235646880
11-4194304-65011713-401604570-1296564975-2426918760-2775432762-1994237820-909761382-261232776-45487413-4369626-177147-10184993248
12167772162852126731971322839728131261515970848165219541863781949782671011361854286435195066610790811091659867391434890753144183951239744
13-67108864-1241513985-9495904212-39543899112-99769893480-161143257897-172027337976-123629132952-60258953376-19836704547-4322622996-595213920-46766808-1594323-691979904448
142684354565368709121450300476972089098947705972856477661113034313403139730027499912057672218047229454887883017557624238693376831916882463394210344347815146068547829695703741715072
Technical details
Node id
n-67981840a9da4671 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-52999a0df2a941cc

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): no
  • 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,       -3;
  2:      16,       33,         9;
  3:     -64,     -225,      -162,       -27;
  4:     256,     1281,      1647,       675,        81;
  5:   -1024,    -6657,    -12780,     -9180,     -2592,      -243;
  6:    4096,    32769,     84969,     91422,     44226,      9477,       729;
  7:  -16384,  -155649,   -511974,   -751977,   -540108,   -194643,    -33534,    -2187;
  8:   65536,   720897,   2883555,   5439933,   5328099,   2820987,    805545,   115911,    6561;
  9: -262144, -3276801, -15466464, -35914266, -45311724, -32551308, -13524408, -3188646, -393660, -19683;
%F Formula
The row polynomials satisfy P(n,x) = -(4-6*x)*P(n-2,x) - (5+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = -4 - 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -4 - 3*x;
P[n_, x_] := P[n, x] = Expand[-(4 - 6*x)*P[n - 2, x] - (5 + 3*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 - 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 - 6*x)*v[i - 2] - (5 + 3*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

  • 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.