Real-rooted atlas

A099040 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-8\,x - 3\,x^{2} + 16\,n + 12\,n x + 3\,n x^{2}\right)\,P_{n-1}(x) + \left(-16\,x - 16\,x^{2} - 3\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18210
264406110
3512512160161200
44096563227525764413100
5327685734438400124161952120143000
62621445570564751362094085056063683281561000
720971525242880543948830474249994241925122022489617040000

All 15 rows

Technical details
Node id
n-c1f0668fd5bd5aaf · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1c406f8a3bd5550a

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:         8,         2;
  2:        64,        40,         6;
  3:       512,       512,       160,        16;
  4:      4096,      5632,      2752,       576,        44;
  5:     32768,     57344,     38400,     12416,      1952,      120;
  6:    262144,    557056,    475136,    209408,     50560,     6368,      328;
  7:   2097152,   5242880,   5439488,   3047424,    999424,   192512,    20224,     896;
  8:  16777216,  48234496,  58982400,  40173568,  16715776,  4362240,   698880,   62976,   2448;
  9: 134217728, 436207616, 614465536, 493355008, 249331712, 82419712, 17854464, 2448384, 193152, 6688;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (16*n+(12*n-8)*x+(3*n-3)*x^2)*P(n-1,x) - (16*x+16*x^2+3*x^3)*d/dx 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[((16*n + (12*n - 8)*x + (3*n - 3)*x^2)*P[n - 1, x] - (16*x + 16*x^2 + 3*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] = ((16*n + (12*n - 8)*x + (3*n - 3)*x^2)*v[i - 1] - (16*x + 16*x^2 + 3*x^3)*deriv(v[i - 1], x))/(2*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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741