Real-rooted atlas

A013620 → basis A008288 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-100 + 60\,x - 9\,x^{2}\right)\,P_{n-4}(x) + \left(140 + 18\,x - 18\,x^{2}\right)\,P_{n-3}(x) + \left(-69 - 36\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(14 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 3\,x\)
\(P_{2}(x) = 64 + 57\,x + 9\,x^{2}\)
\(P_{3}(x) = 458 + 711\,x + 270\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18311
264579130
3458711270271466
430167089469811078115991
5187826110761200248134212243170357
6112564476397661230401139115668153097291783036
7656378345553562664845237109223738249790753946218718406928

All 15 rows

Technical details
Node id
n-78e123f521bb0541 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ccafbaeab3dde430

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,         3;
  2:       64,        57,         9;
  3:      458,       711,       270,        27;
  4:     3016,      7089,      4698,      1107,        81;
  5:    18782,     61107,     61200,     24813,      4212,       243;
  6:   112564,    476397,    661230,    401139,    115668,     15309,      729;
  7:   656378,   3455535,   6266484,   5237109,   2237382,    497907,    53946,    2187;
  8:  3750256,  23757177,  53933994,  58730859,  34323588,  11222469,  2028078,  185895,   6561;
  9: 21094262, 156778443, 431423496, 587909421, 445630410, 198030663, 52216812, 7932249, 629856, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(100-60*x+9*x^2)*P(n-4,x) + (140+18*x-18*x^2)*P(n-3,x) - (69+36*x+9*x^2)*P(n-2,x) + (14+6*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8 + 3*x, P(2,x) = 64 + 57*x + 9*x^2, P(3,x) = 458 + 711*x + 270*x^2 + 27*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 3*x;
P[2, x_] := 64 + 57*x + 9*x^2;
P[3, x_] := 458 + 711*x + 270*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[-(100 - 60*x + 9*x^2)*P[n - 4, x] + (140 + 18*x - 18*x^2)*P[n - 3, x] - (69 + 36*x + 9*x^2)*P[n - 2, x] + (14 + 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 + 3*x; v[3] = 64 + 57*x + 9*x^2; v[4] = 458 + 711*x + 270*x^2 + 27*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(100 - 60*x + 9*x^2)*v[i - 4] + (140 + 18*x - 18*x^2)*v[i - 3] - (69 + 36*x + 9*x^2)*v[i - 2] + (14 + 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

Proof chains ? JSON

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.