Real-rooted atlas

basis-map coefficients of A112227 → 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

\(P_{n}(x) = \left(-4 + 6\,x\right)\,P_{n-2}(x) + \left(5 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 16 + 33\,x + 9\,x^{2}\)
\(P_{2}(x) = 64 + 225\,x + 162\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11633958
26422516227478
325612811647675813940
410246657127809180259224332476
54096327698496991422442269477729267688
6163841556495119747519775401081946433353421872206456
7655367208972883555543993353280992820987805545115911656118187024
826214432768011546646435914266453117243255130813524408318864639366019683149909104

All 15 rows

Technical details
Node id
n-b1ddde2321926af9 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-52999a0df2a941cc

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:      16,       33,         9;
   2:      64,      225,       162,         27;
   3:     256,     1281,      1647,        675,         81;
   4:    1024,     6657,     12780,       9180,       2592,        243;
   5:    4096,    32769,     84969,      91422,      44226,       9477,        729;
   6:   16384,   155649,    511974,     751977,     540108,     194643,      33534,      2187;
   7:   65536,   720897,   2883555,    5439933,    5328099,    2820987,     805545,    115911,      6561;
   8:  262144,  3276801,  15466464,   35914266,   45311724,   32551308,   13524408,   3188646,    393660,    19683;
   9: 1048576, 14680065,  79953885,  221512320,  345628620,  319376358,  178979706,  60886080,  12203460,  1318761,   59049;
  10: 4194304, 65011713, 401604570, 1296564975, 2426918760, 2775432762, 1994237820, 909761382, 261232776, 45487413, 4369626, 177147;
%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 >= 3, with P(1,x) = 16 + 33*x + 9*x^2, P(2,x) = 64 + 225*x + 162*x^2 + 27*x^3.
%t Mathematica
P[1, x_] := 16 + 33*x + 9*x^2;
P[2, x_] := 64 + 225*x + 162*x^2 + 27*x^3;
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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 16 + 33*x + 9*x^2; v[2] = 64 + 225*x + 162*x^2 + 27*x^3; for(n = 3, N, my(i = n); 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, 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.