Real-rooted atlas

A225470 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
24181840
3872198162440
4162401260270019446160
5327206120243004438829160104720
6642016252001587605261768573045248802094400
7128537692736846720438631212764304190531441102248048171200
8256138243144963919104290939041308000963444495844793204162645395201252451200
951234560100224016329600164003616104713560042205017601025877960013470730272714256704036321084800
101024844803041280627264008178952327014379680398550715201471326120003347048485444184070422402142770112001162274713600
1120482027528870400225815040370576166441019302016311184672480160948258416055247858930881188207408643214235986416320707114136960040679614976000
1240964792322503987277096448015540864768215446023936209906421753614421613377600689909448159362226564403240324562266399441925266377937267202545610893056001545825369088000
138192111820868769792251848396861149344256103698577382412600523368576110706731239104700424811358272313595972846956895960544525256321884611454527692821047996133953280992788248291840063378840132608000
1416384258048018450432079252992002280750151684642244006400687544492377607506354439584006050532334758912356897615339865601509024175007731204407265160966304008336328031695375369038716025918745604169710642825728002788668965834752000
Technical details
Node id
n-32d1e8cbf149a672 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9e437d53009ce099

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,     3;
  2:   4,    18,      18;
  3:   8,    72,     198,      162;
  4:  16,   240,    1260,     2700,      1944;
  5:  32,   720,    6120,    24300,     44388,      29160;
  6:  64,  2016,   25200,   158760,    526176,     857304,     524880;
  7: 128,  5376,   92736,   846720,   4386312,   12764304,   19053144,    11022480;
  8: 256, 13824,  314496,  3919104,  29093904,  130800096,  344449584,   479320416,   264539520;
  9: 512, 34560, 1002240, 16329600, 164003616, 1047135600, 4220501760, 10258779600, 13470730272, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (2+3*n*x)*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[(2 + 3*n*x)*P[n - 1, 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] = (2 + 3*n*x)*v[i - 1]); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.