Real-rooted atlas

A249252 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-4 + 4\,x + 44\,x^{2} + 34\,x^{3} + 4\,n - 12\,n x - 64\,n x^{2} - 51\,n x^{3} + 8\,n^{2} x + 20\,n^{2} x^{2} + 17\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 54\,x^{2} + 118\,x^{3} + 68\,x^{4} - 4\,n x - 28\,n x^{2} - 57\,n x^{3} - 34\,n x^{4}\right)\,P_{n-1}'(x) + \left(4\,x^{2} + 24\,x^{3} + 37\,x^{4} + 17\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1459
28241749
3168012461281
4322245525762171601
5645761968321625327739129
612814086176138881696810728275352049
7256332817856513928596883856443009805296761

All 15 rows

Technical details
Node id
n-0eca1578bbac6790 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-205a62c6b3f27a2b

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:    4,     5;
  2:    8,    24,     17;
  3:   16,    80,    124,     61;
  4:   32,   224,    552,    576,     217;
  5:   64,   576,   1968,   3216,    2532,     773;
  6:  128,  1408,   6176,  13888,   16968,   10728,    2753;
  7:  256,  3328,  17856,  51392,   85968,   83856,   44300,    9805;
  8:  512,  7680,  48768, 171520,  366240,  487328,  395560,  179472,  34921;
  9: 1024, 17408, 127744, 531712, 1385536, 2348096, 2592752, 1801936, 716468, 124373;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(8*n^2-12*n+4)*x+(20*n^2-64*n+44)*x^2+(17*n^2-51*n+34)*x^3)*P(n-1,x) - ((4*n-4)*x+(28*n-54)*x^2+(57*n-118)*x^3+(34*n-68)*x^4)*d/dx P(n-1,x) + (4*x^2+24*x^3+37*x^4+17*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (8*n^2 - 12*n + 4)*x + (20*n^2 - 64*n + 44)*x^2 + (17*n^2 - 51*n + 34)*x^3)*P[n - 1, x] - ((4*n - 4)*x + (28*n - 54)*x^2 + (57*n - 118)*x^3 + (34*n - 68)*x^4)*D[P[n - 1, x], x] + (4*x^2 + 24*x^3 + 37*x^4 + 17*x^5)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
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] = 4 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n^2 - 12*n + 4)*x + (20*n^2 - 64*n + 44)*x^2 + (17*n^2 - 51*n + 34)*x^3)*v[i - 1] - ((4*n - 4)*x + (28*n - 54)*x^2 + (57*n - 118)*x^3 + (34*n - 68)*x^4)*deriv(v[i - 1], x) + (4*x^2 + 24*x^3 + 37*x^4 + 17*x^5)*deriv(deriv(v[i - 1], x), x))/(2*n - 2)); 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

Candidate equalities ?

Same fingerprint ?

References

  1. [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