Real-rooted atlas

A125092 → D+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 15\,x^{2} - 20\,x^{3} - 15\,x^{4} - 6\,x^{5} - x^{6}\right)\,P_{n-6}(x) + \left(6 + 30\,x + 60\,x^{2} + 60\,x^{3} + 30\,x^{4} + 6\,x^{5}\right)\,P_{n-5}(x) + \left(-15 - 60\,x - 90\,x^{2} - 60\,x^{3} - 15\,x^{4}\right)\,P_{n-4}(x) + \left(20 + 60\,x + 60\,x^{2} + 20\,x^{3}\right)\,P_{n-3}(x) + \left(-15 - 30\,x - 15\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 16\,x\)
\(P_{2}(x) = 9 + 104\,x + 81\,x^{2}\)
\(P_{3}(x) = 13 + 264\,x + 675\,x^{2} + 256\,x^{3}\)
\(P_{4}(x) = 17 + 496\,x + 2214\,x^{2} + 2624\,x^{3} + 625\,x^{4}\)
\(P_{5}(x) = 21 + 800\,x + 5130\,x^{2} + 10560\,x^{3} + 7625\,x^{4} + 1296\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151621
2910481194
3132646752561208
417496221426246255976
5218005130105607625129625432
62511769855291203637518360240197312
72916241682164960116375101304387594096343968

All 15 rows

Technical details
Node id
n-829b1c502e3aa3c5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1cbc1dd13205f1d1

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,   16;
  2:  9,  104,    81;
  3: 13,  264,   675,    256;
  4: 17,  496,  2214,   2624,    625;
  5: 21,  800,  5130,  10560,   7625,    1296;
  6: 25, 1176,  9855,  29120,  36375,   18360,   2401;
  7: 29, 1624, 16821,  64960, 116375,  101304,  38759,   4096;
  8: 33, 2144, 26460, 126336, 295750,  368928, 242844,  74240,   6561;
  9: 37, 2736, 39204, 223104, 645750, 1052352, 991956, 520704, 131949, 10000;
%F Formula
The row polynomials satisfy P(n,x) = -(1+6*x+15*x^2+20*x^3+15*x^4+6*x^5+x^6)*P(n-6,x) + (6+30*x+60*x^2+60*x^3+30*x^4+6*x^5)*P(n-5,x) - (15+60*x+90*x^2+60*x^3+15*x^4)*P(n-4,x) + (20+60*x+60*x^2+20*x^3)*P(n-3,x) - (15+30*x+15*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 5 + 16*x, P(2,x) = 9 + 104*x + 81*x^2, P(3,x) = 13 + 264*x + 675*x^2 + 256*x^3, P(4,x) = 17 + 496*x + 2214*x^2 + 2624*x^3 + 625*x^4, P(5,x) = 21 + 800*x + 5130*x^2 + 10560*x^3 + 7625*x^4 + 1296*x^5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 16*x;
P[2, x_] := 9 + 104*x + 81*x^2;
P[3, x_] := 13 + 264*x + 675*x^2 + 256*x^3;
P[4, x_] := 17 + 496*x + 2214*x^2 + 2624*x^3 + 625*x^4;
P[5, x_] := 21 + 800*x + 5130*x^2 + 10560*x^3 + 7625*x^4 + 1296*x^5;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6)*P[n - 6, x] + (6 + 30*x + 60*x^2 + 60*x^3 + 30*x^4 + 6*x^5)*P[n - 5, x] - (15 + 60*x + 90*x^2 + 60*x^3 + 15*x^4)*P[n - 4, x] + (20 + 60*x + 60*x^2 + 20*x^3)*P[n - 3, x] - (15 + 30*x + 15*x^2)*P[n - 2, x] + (6 + 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 + 16*x; v[3] = 9 + 104*x + 81*x^2; v[4] = 13 + 264*x + 675*x^2 + 256*x^3; v[5] = 17 + 496*x + 2214*x^2 + 2624*x^3 + 625*x^4; v[6] = 21 + 800*x + 5130*x^2 + 10560*x^3 + 7625*x^4 + 1296*x^5; for(n = 6, N, my(i = n + 1); v[i] = -(1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6)*v[i - 6] + (6 + 30*x + 60*x^2 + 60*x^3 + 30*x^4 + 6*x^5)*v[i - 5] - (15 + 60*x + 90*x^2 + 60*x^3 + 15*x^4)*v[i - 4] + (20 + 60*x + 60*x^2 + 20*x^3)*v[i - 3] - (15 + 30*x + 15*x^2)*v[i - 2] + (6 + 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.