Real-rooted atlas

A038326 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(121 + 264\,x + 144\,x^{2} + 132\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1231235
23855521441081
35687138609936172831211
47852927297633264015897620736863857
51039511471174081892806652800238464024883223226803
61336723374844792169622640196542720119750400343388162985984611452585
7168298295112284757231434812644749433920412739712020118067204807434243583180815839840123
82085127297161566363205389668345610059140044811398641408079246024704321889075206593052672429981696405174228193
925508706839225193748076872458361280194028060441627159678120962462106544128142642844467249662885888089006211072515978035210258739071811
1030889114752130610448206801351162488456034898334451200582084181324806518322749030449242130882560244530590515207449432883200118674948096061917364224257564830589209
11370905171794340773631472772202028958164880594511494920640115164503688960015367022386974721434031004786688928565896642560403475474350080109258348953600156650931486727430083706886421109238115595
1244223308944705534103447383792293570146603958496973899919142402140241381714304033167377062420480368808537287393282950006638418329616714186139566080645560758960128015733202249318402050703103098888916100448256159119229571625809
135241175389124076898836195373980416600688959357761526564762340583683781982096846553606677553110948628488623518036229324808219161688119050245752512944915742722897125597524787201007074783977799682231254137176064026659140340285441069932053790723922717773331637459
146179565473763649880517465372843765795022404114143202332963858172403456641157200183045145612707459845404420096186971487106561597442069644328695037952017260239545050005504107380241638427197444867171003841642496153807785189336678431237557920464896034451812132061184128391846454886496270945129865270153
Technical details
Node id
n-728fca7147e339bb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29d09fbfedb73813

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:          23,           12;
  2:         385,          552,          144;
  3:        5687,        13860,         9936,          1728;
  4:       78529,       272976,       332640,        158976,         20736;
  5:     1039511,      4711740,      8189280,       6652800,       2384640,        248832;
  6:    13367233,     74844792,    169622640,     196542720,     119750400,      34338816,       2985984;
  7:   168298295,   1122847572,   3143481264,    4749433920,    4127397120,    2011806720,     480743424,     35831808;
  8:  2085127297,  16156636320,  53896683456,  100591400448,  113986414080,   79246024704,   32188907520,   6593052672,   429981696;
  9: 25508706839, 225193748076, 872458361280, 1940280604416, 2715967812096, 2462106544128, 1426428444672, 496628858880, 89006211072, 5159780352;
%F Formula
The row polynomials satisfy (12*n-1+12*x)*P(n,x) = (132*n+121+(144*n+264)*x+144*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[((132*n + 121 + (144*n + 264)*x + 144*x^2)*P[n - 1, x])/(12*n - 1 + 12*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] = ((132*n + 121 + (144*n + 264)*x + 144*x^2)*v[i - 1])/(12*n - 1 + 12*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.