Real-rooted atlas

A144697 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2910120
32767251120
481376326561840
524319093134131411916720
6729909425215204204775246160480
7218741479180639248595115105162935011604800
8656118341211935482575404204871059076453388101216652800
919683792697743503623807260297933941496966228435481702842035179833600
105904933676184430945320232384837520457829851766010044024213077032533237408211037836800
111771471412001125522344516127281654245022830501457332626950681386329715305817373516486958177114529715200
12531441586058081431437346122286034404422971329574076769248593576265242250012412838047687352524792805052386163681217945728000
1315943232413319657860456426891174314464318940274659906815736391126832792700640504626948176853200853220535950351075598138153863103275113487131648000
1447829699876483824243959767562903006578840035499381611224439286376219057629297051154197086110806428917598019132576234553812425075096945178008924665019565518159281238016000
Technical details
Node id
n-564851b29dde9cba · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-623afbe97ed5c918

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:     3,      1;
  2:     9,     10,       1;
  3:    27,     67,      25,        1;
  4:    81,    376,     326,       56,        1;
  5:   243,   1909,    3134,     1314,      119,        1;
  6:   729,   9094,   25215,    20420,     4775,      246,       1;
  7:  2187,  41479,  180639,   248595,   115105,    16293,     501,      1;
  8:  6561, 183412, 1193548,  2575404,  2048710,   590764,   53388,   1012,    1;
  9: 19683, 792697, 7435036, 23807260, 29793394, 14969662, 2843548, 170284, 2035, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+n*x)*P(n-1,x) + (x-x^2)*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[(3 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
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] = (3 + n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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