Real-rooted atlas

A196347 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2816630
34814410824324
4384153617287681204536
53840192002880019200600072077760
6460802764805184004608002160005184050401574640
764512045158401016064011289600705600025401604939204032036741600
81032192082575360216760320289013760225792000108380160316108805160960362880969978240
918579456016721510405016453120780337152073156608004389396480170698752041803776058786560362880028570268160
103715891200371589120001254113280002229534720002438553600001755758592008534937600027869184000587865600072576000039916800928533715200
118174960640089924567040033721712640006744342528000843042816000070815596544004130909798400168608563200047421158400087816960000965986560047900160032998659724800
121961990553600235438866432009711853240320021581896089600030349541376000029135559720960019828367032320097118532403200341432340480008430428160000139102064640013795246080062270208001272805446528000
13510117543936006631528071168002984187632025600729468087828480011397938872320000123097739821056009574268652748800547101065871360023080826216448007123711795200001567216594944002331396587520021047330304008717829120052948706575564800
1414283291230208001999660772229120097483462646169600259955900389785600446799203794944000536159044553932800469139163984691200306376596887961600150794731280793600558499004743680001535872263045120030463582076928004125276739584003417389015040013076743680002362836030934579200
Technical details
Node id
n-4f58ee6499244b82 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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,          2;
  2:         8,         16,          6;
  3:        48,        144,        108,         24;
  4:       384,       1536,       1728,        768,        120;
  5:      3840,      19200,      28800,      19200,       6000,        720;
  6:     46080,     276480,     518400,     460800,     216000,      51840,       5040;
  7:    645120,    4515840,   10160640,   11289600,    7056000,    2540160,     493920,     40320;
  8:  10321920,   82575360,  216760320,  289013760,  225792000,  108380160,   31610880,   5160960,   362880;
  9: 185794560, 1672151040, 5016453120, 7803371520, 7315660800, 4389396480, 1706987520, 418037760, 58786560, 3628800;
%F Formula
The row polynomials satisfy (2+n*x)*P(n,x) = (4*n+(2*n^2+4*n)*x+(n^2+n)*x^2)*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[Together[((4*n + (2*n^2 + 4*n)*x + (n^2 + n)*x^2)*P[n - 1, x])/(2 + n*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^2 + 4*n)*x + (n^2 + n)*x^2)*v[i - 1])/(2 + n*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. [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.