Real-rooted atlas

A013609 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(9 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
28110836225
372914589722163375
4656117496174967776129650625
559049196830262440174960583207776759375
653144121257643542940314928015746404199044665611390625
7478296922320522446410444960116033067440132269762939328279936170859375
8430467212295825125356925287142567045952139203174474241058158082015539216796162562890625
93874204892324522934619872782496424655049642465504642831033628570268168162933761360488961007769638443359375
10348678440123245229340697356880201239745564801446369825601157095860486428310336024488801280612220032090699264060466176576650390625
113138105960923012777046676709256822015341851364402045580181920190920816979212728054465286060978316802020326105604489613568059861514243627970568649755859375
1228242953648122594362918488284599736776184102216372802761533245592029456354619648229104980375041309171316428854548804851201616260884480323252176896391820820482176782336129746337890625
132541865828329220295038455188811801538207221539959315617635899932192696043079918631235238293261005542425528840670361612764420335180847275630871040126068348989442292151799808254683533312130606940161946195068359375
14228767924549612135167295796369252391615117562467304430698016452339145627969660311886083729286031188608372928459519132066508826805282703879681191345897950208397115299316736962703755919361604506259865616456474460167836416409629192926025390625
Technical details
Node id
n-5c2409b346bb76f2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f06e51c139cfae05

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:         9,          6;
  2:        81,        108,         36;
  3:       729,       1458,        972,        216;
  4:      6561,      17496,      17496,       7776,       1296;
  5:     59049,     196830,     262440,     174960,      58320,       7776;
  6:    531441,    2125764,    3542940,    3149280,    1574640,     419904,      46656;
  7:   4782969,   22320522,   44641044,   49601160,   33067440,   13226976,    2939328,    279936;
  8:  43046721,  229582512,  535692528,  714256704,  595213920,  317447424,  105815808,  20155392,   1679616;
  9: 387420489, 2324522934, 6198727824, 9642465504, 9642465504, 6428310336, 2857026816, 816293376, 136048896, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (9+6*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[(9 + 6*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] = (9 + 6*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.