Real-rooted atlas

A099040 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4\,x - 2\,x^{2} + 6\,n + 6\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-6\,x - 8\,x^{2} - 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
23630672
321628812016640
4129623761548432445696
5777618144162006984146412050688
64665613219215033688344284404776328451072
72799369331201290816964224421632108288151688964014080

All 15 rows

Technical details
Node id
n-fdf21b802047fbf5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3f5c634f6f648d33

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:        6,        2;
  2:       36,       30,        6;
  3:      216,      288,      120,       16;
  4:     1296,     2376,     1548,      432,       44;
  5:     7776,    18144,    16200,     6984,     1464,      120;
  6:    46656,   132192,   150336,    88344,    28440,     4776,     328;
  7:   279936,   933120,  1290816,   964224,   421632,   108288,   15168,     896;
  8:  1679616,  6438528, 10497600,  9533376,  5288976,  1840320,  393120,   47232,   2448;
  9: 10077696, 43670016, 82021248, 87806592, 59167584, 26078112, 7532352, 1377216, 144864, 6688;
%F Formula
The row polynomials satisfy n*P(n,x) = (6*n+(6*n-4)*x+(2*n-2)*x^2)*P(n-1,x) - (6*x+8*x^2+2*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[((6*n + (6*n - 4)*x + (2*n - 2)*x^2)*P[n - 1, x] - (6*x + 8*x^2 + 2*x^3)*D[P[n - 1, x], x])/(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] = ((6*n + (6*n - 4)*x + (2*n - 2)*x^2)*v[i - 1] - (6*x + 8*x^2 + 2*x^3)*deriv(v[i - 1], 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
  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