Real-rooted atlas

A281000 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n^{2}\right)\,P_{n}(x) = \left(6\,x + n - 36\,n x + 2\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 4\,x^{2} + 4\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2134
31101021
41217035127
5136252420126835
6155660231023104625798
7178143085801801812012171641835
8110527302502590090126126600606435310572
91136476061880340340816816816816291720243102356779
1011717752135660105814838798766651216498841213856709237818199284
111210119702713202848860148140723879876049884120290990706466460352716142547559
121253177105047356864396480507721784742963505745103505745101636014382974571613520781129760415
131300253008855001514205013728792068643960019122246002921454250233716340089237148013520780052003009043402501
141351351001480050310810503543239702294669520860501070018644189850227873431501491535188047457937806084351002005830073007772802
15140647502237510060090030841260420685026342033272708040970453984501682120239801682120239809175201308024702465060271455660077558760593742784829
Technical details
Node id
n-3af5b0030bb79fc0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c613f7c62d85a96e

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   3;
   3: 1,  10,   10;
   4: 1,  21,   70,     35;
   5: 1,  36,  252,    420,     126;
   6: 1,  55,  660,   2310,    2310,     462;
   7: 1,  78, 1430,   8580,   18018,   12012,    1716;
   8: 1, 105, 2730,  25025,   90090,  126126,   60060,    6435;
   9: 1, 136, 4760,  61880,  340340,  816816,  816816,  291720,   24310;
  10: 1, 171, 7752, 135660, 1058148, 3879876, 6651216, 4988412, 1385670, 92378;
%F Formula
The row polynomials satisfy (2*n^2+n)*P(n,x) = (2*n^2+n+(24*n^2-36*n+6)*x)*P(n-1,x) + ((4*n-1)*x-(16*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (24*n^2 - 36*n + 6)*x)*P[n - 1, x] + ((4*n - 1)*x - (16*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 + n + (24*n^2 - 36*n + 6)*x)*v[i - 1] + ((4*n - 1)*x - (16*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.