Real-rooted atlas

A038323 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(47 + 81\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14781128
222097614656116384
31038235367879251015314412097152
4487968133638652869594949991090843046721268435456
5229345007197627080568118270301173953169010115979435348678440134359738368
610779215329111461673402480233805615110351597886014263531003359832732010822824295364814398046511104
7506623120463611181509154331599384409467907641892612351564233900034051617484415779899291931750224922876792454961562949953421312
823811286661761328291782060024198022808965993268254670324448721470379866032007020272471344441288174688316904228128601673963065336185302018885184172057594037927936
911191304731027671735842797642376911966235456087874848119542578736347612439413666630779222143813844674666206246310526834961649218192540431883185647838275398843287431500946352969991219223372036854775808
10525991322358300499064956832132412707030163330451626445323088357314372619609744207372194110389020151850139941862336428941486903107993781028501646676616990836018419947187281725460570544478589589586870121576654590569288011180591620717411303424
112472159215084012303468658268221245736594038438268714989857852087958509144133054165719679315917765010159017364177537249904713198299254974578136655695543683837832952746065553031743708986082173543595182357477154089082058956285513042332432190117984770902183611232881151115727451828646838272
121161914831089485782412402938757061659958516227767918355525427982741308453999063656713943405073739177220243321620951399056590144135179749096281299676103448456353807641553690755706975619437844475862967037586469646895342832057049687474270826017725146779377458776129945554107888315567353448847976644307687250986336119342813113834066795298816
13546099970612058317732712234963171372285288777312651472555929639681586747994653934278942522194174344450515253507629945600051068529470722583243533372072455344315702957240459663484231550841956160350579420045469504545719930665908505695236759671433976169486867153610021560733209104071797781696779959880405826478620624873729671996910352651357164610818892266732989322412475880078570760549798248448
1425666698618766740933436961927736667407413230888186937224118168085758736729147822566261414037996397877222664843903680802050420483297812137685949553047166209134201952069966568233027807321233977657791438790729544654837659982233184728572969463279067689157572760711263904471716945938485467406128976198107348137216040453376901127431078932160345706208698350602229648594251391883111151030697414578523347633027360537213511521316912650057057350374175801344
Technical details
Node id
n-3d15e6fe2cfdd18f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-853215eedfc7a804

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:               47,                81;
  2:             2209,              7614,               6561;
  3:           103823,            536787,             925101,             531441;
  4:          4879681,          33638652,           86959494,           99910908,            43046721;
  5:        229345007,        1976270805,         6811827030,        11739531690,         10115979435,          3486784401;
  6:      10779215329,      111461673402,       480233805615,      1103515978860,       1426353100335,        983273201082,        282429536481;
  7:     506623120463,     6111815091543,     31599384409467,     90764189261235,     156423390003405,     161748441577989,      92919317502249,      22876792454961;
  8:   23811286661761,   328291782060024,   1980228089659932,   6825467032444872,   14703798660320070,   20272471344441288,   17468831690422812,    8601673963065336,   1853020188851841;
  9: 1119130473102767, 17358427976423769, 119662354560878748, 481195425787363476, 1243941366663077922, 2143813844674666206, 2463105268349616492, 1819254043188318564, 783827539884328743, 150094635296999121;
%F Formula
The row polynomials satisfy P(n,x) = (47+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[(47 + 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] = (47 + 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

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.