Real-rooted atlas

basis-map coefficients of A112227 → basis A013609 → basis A008288

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 9 + 16\,x + 4\,x^{2}\)
\(P_{2}(x) = 27 + 82\,x + 52\,x^{2} + 8\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1916429
22782528169
38135239214416985
4243137822801440368325741
57295104114041067245928966433461
6218718226517406573641488134082112128195025
76561634242194083572163049921431043686448642561136689
81968321651488571217739521941408122515245414497024110085126625109

All 15 rows

Technical details
Node id
n-59aeceada33946d1 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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:      9,      16,        4;
   2:     27,      82,       52,        8;
   3:     81,     352,      392,      144,       16;
   4:    243,    1378,     2280,     1440,      368,       32;
   5:    729,    5104,    11404,    10672,     4592,      896,       64;
   6:   2187,   18226,    51740,    65736,    41488,    13408,     2112,      128;
   7:   6561,   63424,   219408,   357216,   304992,   143104,    36864,     4864,     256;
   8:  19683,  216514,   885712,  1773952,  1941408,  1225152,   454144,    97024,   11008,    512;
   9:  59049,  728272,  3444500,  8234400, 11112992,  8964096,  4442496,  1355520,  247040,  24576,  1024;
  10: 177147, 2421010, 13010436, 36276168, 58644448, 58289728, 36786048, 14924288, 3860224, 612864, 54272, 2048;
%F Formula
The row polynomials satisfy P(n,x) = -(3-2*x)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 9 + 16*x + 4*x^2, P(2,x) = 27 + 82*x + 52*x^2 + 8*x^3.
%t Mathematica
P[1, x_] := 9 + 16*x + 4*x^2;
P[2, x_] := 27 + 82*x + 52*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[-(3 - 2*x)*P[n - 2, x] + (4 + 2*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] = 9 + 16*x + 4*x^2; v[2] = 27 + 82*x + 52*x^2 + 8*x^3; for(n = 3, N, my(i = n); v[i] = -(3 - 2*x)*v[i - 2] + (4 + 2*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.