Real-rooted atlas

A081579 → basis A013609 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 4\,x - 16\,n - 2\,n x + 5\,n^{2}\right)\,P_{n-1}(x) + \left(86 + 69\,x + 8\,x^{2} - 32\,n - 24\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(48 + 68\,x + 27\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22824456
3160208808456
49761664976224163856
5582412384985635845763232256
63500888960892164454411456140864270656
72099206210567521284784641704963353633281282269056
81259776425164860328964666368213273658368092416768025619027456
97558144286581764662886442489856237373448366592184524824371217408512159545856
10453498881908469763502397443674767362428528641050071042989260854968326215683891210241337824256
11272097280125845094425714647043053295616233113190411960320004186603529945907215638528154419286016204811217862656
12163258777682309611521853241344024567750656212851671041264929996852654735361540259840312905728428933123756032188416409694063661056
139795518464534672097281315167600641925470945281866761175041261689733126095267430421234778112531547750494102323211419648089784324096008192788739219456
14587731271683452921118729212381102081476294213632158376266956812001010647046605287260162674513674247980841369617413013504272705126429661593621151744884736163846613708537856

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(12 + 8\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-c2bf05ed6c0ffb73 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-4efbef61dc598644)
Fingerprint
f6-64dd59ca82c1978f

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,        2;
  2:      28,       24,        4;
  3:     160,      208,       80,        8;
  4:     976,     1664,      976,      224,       16;
  5:    5824,    12384,     9856,     3584,      576,      32;
  6:   35008,    88960,    89216,    44544,    11456,    1408,      64;
  7:  209920,   621056,   752128,   478464,   170496,   33536,    3328,    128;
  8: 1259776,  4251648,  6032896,  4666368,  2132736,  583680,   92416,   7680,   256;
  9: 7558144, 28658176, 46628864, 42489856, 23737344, 8366592, 1845248, 243712, 17408, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n^2-16*n+15-(2*n-4)*x)*P(n-1,x) - (32*n-86+(24*n-69)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (48+68*x+27*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 16*n + 15 - (2*n - 4)*x)*P[n - 1, x] - (32*n - 86 + (24*n - 69)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (48 + 68*x + 27*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^2 - 16*n + 15 - (2*n - 4)*x)*v[i - 1] - (32*n - 86 + (24*n - 69)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (48 + 68*x + 27*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(n)); 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. [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