Real-rooted atlas

A130123 → 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

\(P_{n}(x) = \left(-16 + 24\,x\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18614
26413236232
3512180012962163824
44096204962635210800129663040
5327682130244089602937608294477761039232
626214420972165438016585100828304646065284665617132032
7209715219923072655326729625305669133824249143044292352279936282426368

All 15 rows

Technical details
Node id
n-803c5f256ddb7d68 · 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): 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:         8,          6;
  2:        64,        132,         36;
  3:       512,       1800,       1296,         216;
  4:      4096,      20496,      26352,       10800,        1296;
  5:     32768,     213024,     408960,      293760,       82944,        7776;
  6:    262144,    2097216,    5438016,     5851008,     2830464,      606528,      46656;
  7:   2097152,   19923072,   65532672,    96253056,    69133824,    24914304,    4292352,     279936;
  8:  16777216,  184549632,  738190080,  1392622848,  1363993344,   722172672,  206219520,   29673216,   1679616;
  9: 134217728, 1677722112, 7918829568, 18388104192, 23199602688, 16666269696, 6924496896, 1632586752, 201553920, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -(16-24*x)*P(n-2,x) + (10+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(16 - 24*x)*P[n - 2, x] + (10 + 6*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] = 8 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 - 24*x)*v[i - 2] + (10 + 6*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.