Real-rooted atlas

A038322 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1271643
27298642561849
3196833499220736409679507
453144112597121119744442368655363418801
51434890742515280503884802985984088473601048576147008443
6387420489137749507220407334401612431360716636160169869312167772166321363049
71046035320343391094768771397240327618738176045148078080160526499843170893824268435456271818611107
82824295364811338925209984277703006515232912948920322437996216320115579079884834245653299257982058496429496729611688200277601
97625597484987406698532532649640261511884813329744312729611848661611315270214291030016277389791723527044820107264104367705292868719476736502592611936843
1020589113209464912200955975979203253588260261120514147280633856053318977250918403791571715620864187238109413376063403380965376014089640214528018554258718720109951162777621611482313284249
115559060566555523362368392486582241073684125886169601908771779353190402262248034788966401876827999232327681112194369915453444707701036679168013948743812382720275530741972992032655495344947217592186044416929293739471222707
121500946352969991211067339628778660416347873656787118950468715784056714854409162104540895313920868703245358963097660058495975434485763050590271768100864112984824880300032029757320133083136052901902458814464569986827839078428147497671065639959630797262576401
134052555153018976267312196841417758171681110033213929806832642411924020390691389443573220770949172428803811435489012450590723011504583911072071681784595308984339005447931534706597062246426112048416780451840618952258768129228810003268828575825929879771682544025645035996273704961718264124282290785243
14109418989131512359209907772354276250683808349660462387889152281682882479973425576837121350677451418787178086416008029053852292481024142293591589798155386889636814668515430629376499686686515614921523219740708603086021591045849098845358821212161260411872400554065921867276848000820838417023606591460474887205759403792793673885357344138503765449
Technical details
Node id
n-f3597891e233edcd · revision 0 · JSON
Definition source
derived:basis_map_A013609
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:            27,             16;
  2:           729,            864,            256;
  3:         19683,          34992,          20736,            4096;
  4:        531441,        1259712,        1119744,          442368,           65536;
  5:      14348907,       42515280,       50388480,        29859840,         8847360,        1048576;
  6:     387420489,     1377495072,     2040733440,      1612431360,       716636160,      169869312,       16777216;
  7:   10460353203,    43391094768,    77139724032,     76187381760,     45148078080,    16052649984,     3170893824,     268435456;
  8:  282429536481,  1338925209984,  2777030065152,   3291294892032,   2437996216320,  1155790798848,   342456532992,   57982058496,    4294967296;
  9: 7625597484987, 40669853253264, 96402615118848, 133297443127296, 118486616113152, 70214291030016, 27738979172352, 7044820107264, 1043677052928, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (27+16*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[(27 + 16*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] = (27 + 16*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.