Real-rooted atlas

A088996 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 24\,n x - 16\,n x^{2} + 12\,n^{2} + 40\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
234096139
315272120015363023
4105238414928363523072084489
59452565620587272524811852807372802880281
610395327576318820814565632347955204224614420643840115777315
7135135484060855204896308379648962138880171832934416484106246606028805358042015
8202702581257760106042262469811036162676726912062721073152888771215367019377459223781703680280465753105
934459425152732772022415103648169579047424769193237760222260426956841451961384964849952882688324713683353695126814720016378907447465
106547290753177598068051752663251244174740956162309362388864079150756534272182480259211264281643785388032279943479558144162391287398400418557984768001055526421893435
111374931057572491175288012965274007632123128518448640727927176920320288150922853990478951576121958401509066967062937619820180447428608170963427026534408738710078095360200907832688640074396407696868975
12316234143225179929041314403503949483828963661733460807168241300361760857601082033176470282243429531418787389447801911914363617281270760312504844288145029217269055488011036772264962949125037994383350169601044720729980928005692509347710483225
13790585358062548271095418140010163320231824720115837386274726656841263579354609920421079719856345702415138163150888185856398859950747026718727741019251164826828810967901126198820864011054864294030711193675203584988710699008309925035178249420805850436087893196800469887081635195664825
1421345804667687513918651363927800314957366410505040388647067339386547230822733681805333760170224436893843126272683835822046793543680204576430867381603532846028928875207715717127777967504798462771200974104045312676567449687840138906848597114885399148417629722509312202704814037117435904035102616527359180800041618000320850821419475
Technical details
Node id
n-3c06e6c930d5be6a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-008012f71849b0c3

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        1,          8;
  2:        3,         40,          96;
  3:       15,        272,        1200,         1536;
  4:      105,       2384,       14928,        36352,        30720;
  5:      945,      25656,      205872,       725248,      1185280,        737280;
  6:    10395,     327576,     3188208,     14565632,     34795520,      42246144,      20643840;
  7:   135135,    4840608,    55204896,    308379648,    962138880,    1718329344,    1648410624,     660602880;
  8:  2027025,   81257760,  1060422624,   6981103616,  26767269120,   62721073152,   88877121536,   70193774592,   23781703680;
  9: 34459425, 1527327720, 22415103648, 169579047424, 769193237760, 2222604269568, 4145196138496, 4849952882688, 3247136833536, 951268147200;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(16*n^3-40*n^2+24*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.