Real-rooted atlas

A038221 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 24\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161824
236216324576
321619445832583213824
412961555269984139968104976331776
577761166406998402099520314928018895687962624
646656839808629856025194240566870406802444834012224191102976
7279936587865652907904264539520793618560142851340814285134086122200324586471424
81679616403107844232632322539579392952342272022856214528342843217922938656153611019960576110075314176
9100776962720977923265173504228562145281028529653763085588961286171177922567934371614725950778711041983592903682641807540224
1060466176181398528024488801280195910410240102852965376037027067535369256766883840158687432294401785233613312011901557422080357046722662463403380965376
11362797056119723028481795845427201616260884480969756530688040729774288896122189322866688261834263285760392751394928640392751394928640235650836957184642684100792321521681143169024
1221767823367836416409612930087075841293008707584087278087761920418934821257216146627187440025637704133913149447069525108715520942603347828736084834301304586244627325525704704115683138142617636520347436056576
13130606940165093670666249168607199232100854679191552756410093936640408461450725785616338458029031424490153740870942721102845916959621121838076528266035202205691833919242241804656955024834569023284775124172820822964865671168876488338465357824
147836416409632912948920326418025039462477016300473548863538447890677763812306873440665617155380930482995258818448904513126415439842837434695683087968567486939136463195285123040870450530394740695367683789779605552152576174912904871637811237481336758208102421035720123168587776
Technical details
Node id
n-cbe7affc500960a4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f9762973ec1089d7

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:        6,        18;
  2:       36,       216,        324;
  3:      216,      1944,       5832,        5832;
  4:     1296,     15552,      69984,      139968,       104976;
  5:     7776,    116640,     699840,     2099520,      3149280,      1889568;
  6:    46656,    839808,    6298560,    25194240,     56687040,     68024448,     34012224;
  7:   279936,   5878656,   52907904,   264539520,    793618560,   1428513408,   1428513408,    612220032;
  8:  1679616,  40310784,  423263232,  2539579392,   9523422720,  22856214528,  34284321792,  29386561536,  11019960576;
  9: 10077696, 272097792, 3265173504, 22856214528, 102852965376, 308558896128, 617117792256, 793437161472, 595077871104, 198359290368;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (6+24*x+18*x^2)*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[Together[((6 + 24*x + 18*x^2)*P[n - 1, x])/(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] = ((6 + 24*x + 18*x^2)*v[i - 1])/(1 + x)); 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.