Real-rooted atlas

A038335 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(39 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1393675
21521280812965625
35931916426815163246656421875
423134418541936118272967278336167961631640625
590224199416419380768774240709637760327525120604661762373046875
6351874376119488426984449732930405535174528038320439040141490851842176782336177978515625
713723100667988672342777224555417999843777756615360348715995264019313501276165942615777287836416409613348388671875
8535200926048139522529923552127688173599168235732012798464271998476305920200860413272064927048061255682444961919795228211099074561001129150390625
92087283611587591734051000395844640264984761542413790322748710144190942930366755841762550126462361610846462316691456429090816924057699020957751705610155995666841675084686279296875
108140406085191601751422100171532403121291800712519207683179817138508801241129047383912960137478909864064204810575300758774169605578180620012748801930908676158259203960838310068224036561584400629765631351470947265625
1131747583732247243932236008097358739961487815758339634152041201051769405253440760634801896712371209829742055280590643290736080510282375168598259871496367308802761199406906310656084959981750963404801568491970787016704131621703842267136422351360321044921875
1212381557655576425121137149561723308093648696297774902948783136214245469200907317888044497135910957673715206571884688387594887168707741427980202526310455997123972059980103683230603306080383467520132537571531502911488036702712116416190873661598957398181019648473838133832161689631676352024078369140625
13482880748567480579719579456898280976695662832092997443254093913632108622452884860010169216250667198965061561928960416493192126563825974272512607005694232401199104473175697563906831876096327583175236550883606528167991371916179940311040620275834767433625763841561533570043888848076824023593385290597662721705817281795782082562375726401805877685546875
14188323491941317426090412433718972780102121783761460231383668061273070256539162357046668777749017613686429063492361281321216252672536556782054424391683498542813863136138183884836907704409984732886335488298100689465261304081940481834465781324684948196556884667651445754689916764162841991097479877703499776655844099418433316192256931376235860497017077766140942214464815497216178179480135440826416015625
Technical details
Node id
n-232fd19b6a19d182 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-42b2d489dcb16cdd

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:              39,               36;
  2:            1521,             2808,             1296;
  3:           59319,           164268,           151632,             46656;
  4:         2313441,          8541936,         11827296,           7278336,           1679616;
  5:        90224199,        416419380,        768774240,         709637760,         327525120,          60466176;
  6:      3518743761,      19488426984,      44973293040,       55351745280,       38320439040,       14149085184,        2176782336;
  7:    137231006679,     886723427772,    2455541799984,     3777756615360,     3487159952640,     1931350127616,      594261577728,      78364164096;
  8:   5352009260481,   39522529923552,  127688173599168,   235732012798464,   271998476305920,   200860413272064,    92704806125568,   24449619197952,   2821109907456;
  9: 208728361158759, 1734051000395844, 6402649847615424, 13790322748710144, 19094293036675584, 17625501264623616, 10846462316691456, 4290908169240576, 990209577517056, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (39+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[(39 + 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] = (39 + 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.