Real-rooted atlas

A145324 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2708224176
391016269681923696
414560351163174812752192096096
52766408419241024604623264189504230402978976
66086080223952883432822428056264128947843159936322560107243136
7152152000657259480121653020812506581847712698242853149445862297651609604396968576
8426025600021142001440458935164645691597289644107402384218716752646777112320119972044892897280202260554496
9132067936000740607164640184553903918426822654890562505648931824156016998086464752398720017273358028826874224640185794560010315288279296
10449030982400028086138189760790416849543361317988854899521442019044412481081700558495045633955514380820118469448192471386240409665440309248040874803200577656143640576
1116614146348800011469545487971203598609659864672677355920203228884986437160850247463137773021600468064488070899220965326930344966572561757081601373456121200641721804193792098099527680035237024762075136
1266456585395200005019786000257280017376520466331200036450621923777299251605828789624044851949024753907462438126737732692128020555807461981363280800097047223296225824850532147204259707592638464486908901457920255058771968002325643634296958976
132857633171993600002344586419217190400887744388059445440020539193157796974656323966805181959830403678771270527494137630940224156151704512195144837738057820169230030262385383424323344957461048524881547700797346611214020889535656755214730201960284160714164561510400165120698035084087296
14131451125911705600001164238747999715584004787000110838606144001211126201676494466176210642309857092445952026641351999885267944962526881692341226648832182587297827961710809610100159052839010979844256395883036438241281345154296050939985923091391942560609075248838561508700979204747576286380032002142493684531200012549173050666390634496
Technical details
Node id
n-b3aee49351d9bfdc · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7e79d1acae7aa409

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:            7,            4;
  2:           70,           82,            24;
  3:          910,         1626,           968,           192;
  4:        14560,        35116,         31748,         12752,          1920;
  5:       276640,       841924,       1024604,        623264,        189504,         23040;
  6:      6086080,     22395288,      34328224,      28056264,      12894784,       3159936,       322560;
  7:    152152000,    657259480,    1216530208,    1250658184,     771269824,     285314944,     58622976,      5160960;
  8:   4260256000,  21142001440,   45893516464,   56915972896,   44107402384,   21871675264,   6777112320,   1199720448,    92897280;
  9: 132067936000, 740607164640, 1845539039184, 2682265489056, 2505648931824, 1560169980864, 647523987200, 172733580288, 26874224640, 1857945600;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+4+(2*n+2)*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[(3*n + 4 + (2*n + 2)*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] = (3*n + 4 + (2*n + 2)*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

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.