Real-rooted atlas

A038334 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(44 + 48\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1936 + 4224\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01936422423048464
185184278784304128110592778688
23748096163553282676326419464192530841671639296
31649162248995430401962639360214106112011678515202548039686590815232
47256313856474958725121295341977601884133785601541564006406726824755212230590464606355001344
531927780966424381214556167979306582016145078301491201582672379904010359310123008376702186291258706834227255784660123648
614048223625216122602678910976468119319478272102135124249804813927516943155201215492387766272662995847872512206648056479744281792804290565132188731375616
7618121839509504606883260609331226482178644770816674091820048711681103059341897891841203337463888609288751545191917158440916315182989312111589950499061761352605460594688472161363286556672

All 15 rows

Technical details
Node id
n-49a9936529864dac · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-60cc3cc9b4461a16

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:                1936,                 4224,                 2304;
  1:               85184,               278784,               304128,                110592;
  2:             3748096,             16355328,             26763264,              19464192,               5308416;
  3:           164916224,            899543040,           1962639360,            2141061120,            1167851520,             254803968;
  4:          7256313856,          47495872512,         129534197760,          188413378560,          154156400640,           67268247552,           12230590464;
  5:        319277809664,        2438121455616,        7979306582016,        14507830149120,        15826723799040,        10359310123008,         3767021862912,          587068342272;
  6:      14048223625216,      122602678910976,      468119319478272,      1021351242498048,      1392751694315520,      1215492387766272,       662995847872512,       206648056479744,        28179280429056;
  7:     618121839509504,     6068832606093312,    26482178644770816,     67409182004871168,    110305934189789184,    120333746388860928,     87515451919171584,     40916315182989312,     11158995049906176,      1352605460594688;
  8:   27197360938418176,   296698482964561920,  1456519825462394880,   4237148583163330560,   8089101840584540160,  10589369682219761664,   9626699711108874240,   6001059560171765760,   2454978910979358720,    595146402661662720,    64925062108545024;
  9: 1196683881290399744, 14360206575484796928, 78328399502644346880, 256347489281381498880, 559303612977559633920, 854209154365727440896, 931864532035339026432, 726128206780783656960, 396069930971336540160, 144025429444122378240, 31423730060535791616, 3116402981210161152;
%F Formula
The row polynomials satisfy P(n,x) = (44+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1936 + 4224*x + 2304*x^2.
%t Mathematica
P[0, x_] := 1936 + 4224*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(44 + 48*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] = 1936 + 4224*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (44 + 48*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.