Real-rooted atlas

A075497 → D+1 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-32 + 4\,x^{2} + 4\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(-16\,x + 8\,x^{3} - 2\,x^{4}\right)\,P_{n-4}'(x) + \left(56 - 6\,x^{2}\right)\,P_{n-3}(x) + \left(24\,x - 4\,x^{2} - 2\,x^{3}\right)\,P_{n-3}'(x) + \left(-36 + 2\,x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(10\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 10 + 16\,x + 3\,x^{2}\)
\(P_{3}(x) = 36 + 104\,x + 45\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21016329
336104454189
41366404809651357
5528387245001520175610601
620802329639648201603850288789329
782561399043376802432646860084004418806553

All 15 rows

Technical details
Node id
n-e00b13308d6b02fb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-136022f4f6319ded

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:      3,       2;
  2:     10,      16,        3;
  3:     36,     104,       45,        4;
  4:    136,     640,      480,       96,        5;
  5:    528,    3872,     4500,     1520,      175,       6;
  6:   2080,   23296,    39648,    20160,     3850,     288,      7;
  7:   8256,  139904,   337680,   243264,    68600,    8400,    441,     8;
  8:  32896,  839680,  2818560,  2774016,  1085280,  193536,  16464,   640,   9;
  9: 131328, 5038592, 23227200, 30503680, 15934800, 3884832, 476280, 29760, 891, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(32-4*x^2-4*x^3+x^4)*P(n-4,x) - (16*x-8*x^3+2*x^4)*d/dx P(n-4,x) + (56-6*x^2)*P(n-3,x) + (24*x-4*x^2-2*x^3)*d/dx P(n-3,x) - (36-2*x^2)*P(n-2,x) - (12*x-2*x^2)*d/dx P(n-2,x) + 10*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 10 + 16*x + 3*x^2, P(3,x) = 36 + 104*x + 45*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 10 + 16*x + 3*x^2;
P[3, x_] := 36 + 104*x + 45*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(32 - 4*x^2 - 4*x^3 + x^4)*P[n - 4, x] - (16*x - 8*x^3 + 2*x^4)*D[P[n - 4, x], x] + (56 - 6*x^2)*P[n - 3, x] + (24*x - 4*x^2 - 2*x^3)*D[P[n - 3, x], x] - (36 - 2*x^2)*P[n - 2, x] - (12*x - 2*x^2)*D[P[n - 2, x], x] + 10*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = 3 + 2*x; v[3] = 10 + 16*x + 3*x^2; v[4] = 36 + 104*x + 45*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(32 - 4*x^2 - 4*x^3 + x^4)*v[i - 4] - (16*x - 8*x^3 + 2*x^4)*deriv(v[i - 4], x) + (56 - 6*x^2)*v[i - 3] + (24*x - 4*x^2 - 2*x^3)*deriv(v[i - 3], x) - (36 - 2*x^2)*v[i - 2] - (12*x - 2*x^2)*deriv(v[i - 2], x) + 10*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.