Real-rooted atlas

A113413 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n\right)\,P_{n}(x) = \left(58 + 32\,x - 34\,n - 16\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(116 + 142\,x + 40\,x^{2} - 28\,n - 32\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(35 + 68\,x + 42\,x^{2} + 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
2475616119
3317572336641289
4213751684608179225613961
51440743684522883084889601024151211
6971273541205321764216321856004300840961637759
7654797278927650457125021888296832010414082007041638417738489

All 15 rows

Technical details
Node id
n-76e410d25ce70aab · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3f270cf733a15daf

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:        7,         4;
  2:       47,        56,        16;
  3:      317,       572,       336,        64;
  4:     2137,      5168,      4608,      1792,       256;
  5:    14407,     43684,     52288,     30848,      8960,      1024;
  6:    97127,    354120,    532176,    421632,    185600,     43008,      4096;
  7:   654797,   2789276,   5045712,   5021888,   2968320,   1041408,    200704,    16384;
  8:  4414417,  21513952,  45508736,  54551040,  40512000,  19079168,   5562368,   917504,   65536;
  9: 29760487, 163306948, 395493888, 554633472, 496205312, 293603328, 114860032, 28639232, 4128768, 262144;
%F Formula
The row polynomials satisfy (2*n+2)*P(n,x) = (4*n^2-34*n+58-(16*n-32)*x)*P(n-1,x) - (28*n-116+(32*n-142)*x+(8*n-40)*x^2)*d/dx P(n-1,x) + (35+68*x+42*x^2+8*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[((4*n^2 - 34*n + 58 - (16*n - 32)*x)*P[n - 1, x] - (28*n - 116 + (32*n - 142)*x + (8*n - 40)*x^2)*D[P[n - 1, x], x] + (35 + 68*x + 42*x^2 + 8*x^3)*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; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 34*n + 58 - (16*n - 32)*x)*v[i - 1] - (28*n - 116 + (32*n - 142)*x + (8*n - 40)*x^2)*deriv(v[i - 1], x) + (35 + 68*x + 42*x^2 + 8*x^3)*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 ?

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