Real-rooted atlas

A059364 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(x + 4\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
232441591
33847524881051729
461441548814576607294543225
512288037734446188828177685692103951339975
629491201065369615990784127671685719696136347613513549579075
7825753603425402886075473925973424643516590081239727682423592020270252131900225
82642411520123650703362526470144029443264512214079101449945331712288308649647687544034459425104463111025
99512681472049534835097611444655882241539986849792133010679091276478223436829275241638471946159264103011726606547290755745471106375
1038050725888002181159714816056180938899456856332513443848554399548211258523531983872277705235770889025647114624192291625094424251378886013749310575350473737488875
11167423193907200104722694406144029736280459837445060024653840384573350058213376045425473033789442567944273020928103585051531648029219819867788854897680481552618278680908031623414322523481740411754625
12803631330754560054452473162629120168914819808755712317221884533932032401708644288430080361379785115533312236825007689478144113919431560714240399217764194506249940043630061696166921577887464016974890910180079058535806251714167050058087625
13417888291992371200304850906376044544010253787405446283264210562381306008698882945384038541452902429637882221795344384220721545979722629121231808564877305036851517570259507169281594770232088375040355180398513147072538957693029088804994324931941100213458046676875135419196954588922375
14234017443515727872001828352680383637094406626188575540447805441476509170071590928384226004596737063885209625138827755975606272002095539241918511710208132990527967248644505664572287726765860864023870808674951979315266138439047299112192133183946378441623681842659505973059120156789073640196900619028335362937511510631741140058401875
Technical details
Node id
n-75656e7178d4d4e4 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a039d5598c7c2c03

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:           4,            3;
  2:          32,           44,            15;
  3:         384,          752,           488,           105;
  4:        6144,        15488,         14576,          6072,           945;
  5:      122880,       377344,        461888,        281776,         85692,        10395;
  6:     2949120,     10653696,      15990784,      12767168,       5719696,      1363476,       135135;
  7:    82575360,    342540288,     607547392,     597342464,     351659008,    123972768,     24235920,     2027025;
  8:  2642411520,  12365070336,   25264701440,   29443264512,   21407910144,   9945331712,   2883086496,   476875440,    34459425;
  9: 95126814720, 495348350976, 1144465588224, 1539986849792, 1330106790912, 764782234368, 292752416384, 71946159264, 10301172660, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+(2*n+1)*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[(4*n + (2*n + 1)*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] = (4*n + (2*n + 1)*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.