Real-rooted atlas

delannoy → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(6\,x^{2} + 4\,x^{3} + 2\,n - n x - 10\,n x^{2} - 6\,n x^{3} + 3\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(13\,x^{2} + 22\,x^{3} + 8\,x^{4} - 2\,n x - 10\,n x^{2} - 12\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 9\,x^{3} + 8\,x^{4} + 2\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2410519
3832361288
4168816011629409
532224568656350701900
66454417762856240410141698827
71281280512010624124008192285640841008

All 15 rows

Technical details
Node id
n-96ad2f3514da79b6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f9aae0db5233b03e

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:   2,    2;
  2:   4,   10,     5;
  3:   8,   32,    36,     12;
  4:  16,   88,   160,    116,     29;
  5:  32,  224,   568,    656,    350,     70;
  6:  64,  544,  1776,   2856,   2404,   1014,    169;
  7: 128, 1280,  5120,  10624,  12400,   8192,   2856,   408;
  8: 256, 2944, 13952,  35584,  53536,  48848,  26528,  7880,   985;
  9: 512, 6656, 36480, 110592, 204608, 240192, 179536, 82720, 21402, 2378;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(3*n^2-n)*x+(4*n^2-10*n+6)*x^2+(2*n^2-6*n+4)*x^3)*P(n-1,x) - (2*n*x+(10*n-13)*x^2+(12*n-22)*x^3+(4*n-8)*x^4)*d/dx P(n-1,x) + (2*x^2+9*x^3+8*x^4+2*x^5)*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[((2*n + (3*n^2 - n)*x + (4*n^2 - 10*n + 6)*x^2 + (2*n^2 - 6*n + 4)*x^3)*P[n - 1, x] - (2*n*x + (10*n - 13)*x^2 + (12*n - 22)*x^3 + (4*n - 8)*x^4)*D[P[n - 1, x], x] + (2*x^2 + 9*x^3 + 8*x^4 + 2*x^5)*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] = ((2*n + (3*n^2 - n)*x + (4*n^2 - 10*n + 6)*x^2 + (2*n^2 - 6*n + 4)*x^3)*v[i - 1] - (2*n*x + (10*n - 13)*x^2 + (12*n - 22)*x^3 + (4*n - 8)*x^4)*deriv(v[i - 1], x) + (2*x^2 + 9*x^3 + 8*x^4 + 2*x^5)*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

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