Real-rooted atlas

A038221 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1183654
2324129612962916
35832349926998446656157464
41049768398082519424335923216796168503056
51889568188956807558272015116544015116544060466176459165024
6340122244081466882040733440544195584081629337606530347008217678233624794911296
7612220032857108044851426482688171421608960342843217920411411861504274274574336783641640961338925209984
8110199605761763193692161234235584512493694233804812342355845120197477693521921974776935219211284439629824282110990745672301961339136
919835929036835704672266242856373781299213329744312729639989232938188879978465876377610663795450183689140396100157444570198050078721015599566684163904305912313344
1035704672266247140934453248064268410079232034276485375590401199676988145664028792247715495936479870795258265605484237660094464041131782450708480182807922003148803656158440062976210832519264920576
116426841007923214139050217431041413905021743104084834301304586240339337205218344960950144174611365888190028834922273177627146976417467596802714697641746759680180979842783117312072391937113246924813162170384226713611384956040305711104
1211568313814261762776395315422822430540348469651046420360232313100697609162104540895313920293187345308650045446841038057201834393611727493812346001817614659367265432502272013030548680384446464078183292082306678784284302880299297013764738381338321616896614787626176508399616
1320822964865671168541397086507450368649676503808940441647642943612655632384238214718063278161920857572985027801382912228686129340747035443245737225868149407088646860583880222411063296762287097802490118144060982967824199209451523326343699501775060992110878123316725835366417058172817957820825633198531813531453579264
14374813367582081024104947742922982686721364320657998774927361091456526399019941888600301089519460968038424012043580778438721536720361307423353161646081646540131253378655191042881445229693412646584323841926972924550195445763841926972924550195445762794128707581491051233281397064353790745525616644298659550125370848051261409422144648154972161792720717930698493280256
Technical details
Node id
n-900c1d03ececa54d · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-11aec07ce60b6d61

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:           18,            36;
  2:          324,          1296,           1296;
  3:         5832,         34992,          69984,           46656;
  4:       104976,        839808,        2519424,         3359232,         1679616;
  5:      1889568,      18895680,       75582720,       151165440,       151165440,        60466176;
  6:     34012224,     408146688,     2040733440,      5441955840,      8162933760,      6530347008,       2176782336;
  7:    612220032,    8571080448,    51426482688,    171421608960,    342843217920,    411411861504,     274274574336,     78364164096;
  8:  11019960576,  176319369216,  1234235584512,   4936942338048,  12342355845120,  19747769352192,   19747769352192,  11284439629824,   2821109907456;
  9: 198359290368, 3570467226624, 28563737812992, 133297443127296, 399892329381888, 799784658763776, 1066379545018368, 914039610015744, 457019805007872, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (18+36*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[(18 + 36*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] = (18 + 36*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. [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.