Real-rooted atlas

A038335 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(48 + 81\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14881129
223047776656116641
31105925598729447845314412146689
45308416358318089069926410203667243046721276922881
5254803968214990848072559411201224440064010331213040348678440135723051649
6122305904641238347284485224277606401175462461440148769467776010041939074882824295364814608273662721
7587068342272693474479308835107145515008987388467609601666218039091201687045764579849489632425761622876792454961594467302491009
828179280429056380420285792256224685731296051275831434312417281599569317527552021594185786621952182200942574622728784688302705024185302018885184176686282021340161
913526054605946882054269543278182413866319417127731254598632704940441613820278903438049282332172064955170816262369357307456716818974926733842851848005047215839953121500946352969991219892530380752880769
1064925062108545024109561042308169728083197916502766387203743906242624487424011056223122750439424022388851823569639833631484322876894806016030359882774148562944019212113318015887488072045424942559578080121576654590569288011276136419117121619201
113116402981210161152578482303387136163844880944434828961382402470978120132161699840833955115544604573696019702189604741283053568332474449580009151528964007504526187610308608033813319439707961978880190199921848357286131206419247362382058406928984770902183611232881164621598066108688876929
121495873430980877352963029143697736276639744281142399446148175626241581425996884583487897606004476831921152930611201621208744618711291265024319175471596808785467801646166452141681270755164164869117999317946524958720365183849948845989371904018487432403660328211952645672280396577600701394567976644307687250986336121236186150528020865123841
13718019246870821129420815751547228228638526668815948441568581496508252169868098220559800964481024416310393679866603189043201264542820802594807186718722845221346805838316170117124801311022734852158537072646076659263148797263148482565696868059201997434201702403845385939961348268086149121769751483732211418835102724977426047996844615473726464610818892266732989322412739468013418114691600975489
1434464923849799414212198481423382595151116076318728931127278405638044621209660285109129238056801193164827976058455287035734303703049441919728659374560327499776238998593131690418558289838084609258581825458072195589734468058583747266529347263004672765659067156748455156708802566460248379135065090384730521639642433235601535781906300928167241515212693979079917207044341847029560324456882465952523347633027360537213511521353391373730936795216525838081
Technical details
Node id
n-b0b710f447d722c0 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-50db7ebeb5f2c85c

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:               48,                81;
  2:             2304,              7776,               6561;
  3:           110592,            559872,             944784,             531441;
  4:          5308416,          35831808,           90699264,          102036672,            43046721;
  5:        254803968,        2149908480,         7255941120,        12244400640,         10331213040,          3486784401;
  6:      12230590464,      123834728448,       522427760640,      1175462461440,       1487694677760,       1004193907488,        282429536481;
  7:     587068342272,     6934744793088,     35107145515008,     98738846760960,     166621803909120,     168704576457984,      94896324257616,      22876792454961;
  8:   28179280429056,   380420285792256,   2246857312960512,   7583143431241728,   15995693175275520,   21594185786621952,   18220094257462272,    8784688302705024,   1853020188851841;
  9: 1352605460594688, 20542695432781824, 138663194171277312, 545986327049404416, 1382027890343804928, 2332172064955170816, 2623693573074567168, 1897492673384285184, 800504721583995312, 150094635296999121;
%F Formula
The row polynomials satisfy P(n,x) = (48+81*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[(48 + 81*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] = (48 + 81*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.