Real-rooted atlas

A013620 → basis A008288 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-100 + 60\,x - 9\,x^{2}\right)\,P_{n-4}(x) + \left(140 + 18\,x - 18\,x^{2}\right)\,P_{n-3}(x) + \left(-69 - 36\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(14 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 6\,x\)
\(P_{2}(x) = 25 + 78\,x + 27\,x^{2}\)
\(P_{3}(x) = 125 + 666\,x + 567\,x^{2} + 108\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
2257827130
31256665671081466
462547827047313240515991
53125313146817551300149851458170357
6156251938785691876286683049656561051031783036
778125115650643159236437124453478515965102704591749618406928

All 15 rows

Technical details
Node id
n-3175a393012e8216 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-39e8113cbc48b395

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:       5,        6;
  2:      25,       78,        27;
  3:     125,      666,       567,       108;
  4:     625,     4782,      7047,      3132,       405;
  5:    3125,    31314,     68175,     51300,     14985,      1458;
  6:   15625,   193878,    569187,    628668,    304965,     65610,      5103;
  7:   78125,  1156506,   4315923,   6437124,   4534785,   1596510,    270459,    17496;
  8:  390625,  6719262,  30596319,  58313628,  55190565,  27942570,   7659603,  1067256,   59049;
  9: 1953125, 38282274, 206455959, 483473124, 583801425, 394644150, 154299411, 34484616, 4074381, 196830;
%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) = 5 + 6*x, P(2,x) = 25 + 78*x + 27*x^2, P(3,x) = 125 + 666*x + 567*x^2 + 108*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 6*x;
P[2, x_] := 25 + 78*x + 27*x^2;
P[3, x_] := 125 + 666*x + 567*x^2 + 108*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] = 5 + 6*x; v[3] = 25 + 78*x + 27*x^2; v[4] = 125 + 666*x + 567*x^2 + 108*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.