Real-rooted atlas

A013620 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1143650
2196100812962500
32744211685443246656125000
4384163951361524096261273616796166250000
55378246914880355622409144576011757312060466176312500000
67529536116169984746807040256048128049380710405079158784217678233615625000000
71054135041897443072146374179846273179136016131032064024887878041621332466892878364164096781250000000
8147578905630359089152273231802368140519212646445166889779209291474468864119461814599688776786378752282110990745639062500000000
9206610467844781556541444918172442624295090346557441138205622435842926814457692165017396213186565529375418613763554598483394561015599566684161953125000000000
102892546549767437976842240860680177459205901806931148802655813119016960819508048153804817560886746152960258037519535308802488218938376192014218393933578240365615844006297697656250000000000
11404956516966411454484337049614727194147635201136097834246144058427888618373120210340399026143232540875311781511168993444450210938880127728572169977856010948163328855244805630483997696983041316217038422671364882812500000000000
1256693912375296174941215329484824741686168027136212071595725946880122698566098583552050481695766274375681514450872988231270433379733527087546368536460003113906995206130971464158937088047296065580654657536221124462455008788484738381338321616896244140625000000000000
13793714773254144265327509916385284093624438709944323859703042212233216248123766999357849601148458578682742046723937572269769401303041012518583654988906496195271441133462146252827895920161923163750402869294645226382557184201223260834057997516886238540357453427507217058172817957820825612207031250000000000000
141111200682555801640003224572008857666862532498929090566877289057032706457648632258331874138521625010875713535271239689647052060935033192448283505203423396893818886378867077026430110924810935200703473880190156814059543761609274530201613146586374491789171097684513769550304358957056334340187231973288181766140942214464815497216610351562500000000000000
Technical details
Node id
n-557aaa3e0cf45f29 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-c53984b4362c13dd

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:          14,           36;
  2:         196,         1008,          1296;
  3:        2744,        21168,         54432,          46656;
  4:       38416,       395136,       1524096,        2612736,         1679616;
  5:      537824,      6914880,      35562240,       91445760,       117573120,        60466176;
  6:     7529536,    116169984,     746807040,     2560481280,      4938071040,      5079158784,      2176782336;
  7:   105413504,   1897443072,   14637417984,    62731791360,    161310320640,    248878780416,    213324668928,     78364164096;
  8:  1475789056,  30359089152,  273231802368,  1405192126464,   4516688977920,   9291474468864,  11946181459968,   8776786378752,   2821109907456;
  9: 20661046784, 478155654144, 4918172442624, 29509034655744, 113820562243584, 292681445769216, 501739621318656, 552937541861376, 355459848339456, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (14+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[(14 + 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] = (14 + 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.