Real-rooted atlas

A038326 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 23\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1242347
257611045292209
313824397443808812167103823
43317761271808182822411680322798414879681
57962624381542407312896070081920335809206436343229345007
6191102976109884211226326425603363932160241782624092683339214803588910779215329
74586471424307675791368845679001614128515072013539826944077854004928248700293523404825447506623120463
8110075314176843910742016283061728051254253497876486499116933120498265631539223875228177926537264858247831098528123811286661761
926418075402242278559003443287344761798656195312592355328280761851510784269063441031168171901642881024706024604689921691517282069618011526614631119130473102767
106340338096537660761573425152026203428539596806696431737896960112304740604313601291504516949606410314098572861440564819683751936020298207384835204322766387511204142651121364952599132235830049
111521681143169024160410553842401287686339038281728022098224735059968042354930742198272056826198745782681654458440464708403237278099127627776017862422498654976057060516315147840109365989604033369528097579139272472159215084012303
12365203474360565764199839955146506242213665643025137664707143191521918976015247775067191377920233799217696934461442614005142306003353621472185097513598976128609441990315827205477809566254192640157487025029808038427440921027921097621914624432020321116191483108948578241
138764883384653578241091958388338091622462787607329440268288220628675754838720512528589535662634434560911816949018044399616116509943485639006617611165536250707071467528025229180195707617284272691461678270259201637865060310003599364280783680355691225668373628227903401525040363619364673835460999706120583177327
142103572012316858777628222924498584521932817580530052243275120646739203186693255462912177606083982645170012163404116609667365758566448934176263968382779392535945740033939430440964494128340909596265676828712486622477976141824137580665066040302346244794477721998374172672114867695422877714553616935621761065304068811592836324538749809256666986187667409334369
Technical details
Node id
n-1473f71bb1b53b4e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-723ac5bf98d71a32

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:            24,             23;
  2:           576,           1104,            529;
  3:         13824,          39744,          38088,           12167;
  4:        331776,        1271808,        1828224,         1168032,          279841;
  5:       7962624,       38154240,       73128960,        70081920,        33580920,         6436343;
  6:     191102976,     1098842112,     2632642560,      3363932160,      2417826240,       926833392,       148035889;
  7:    4586471424,    30767579136,    88456790016,    141285150720,    135398269440,     77854004928,     24870029352,     3404825447;
  8:  110075314176,   843910742016,  2830617280512,   5425349787648,   6499116933120,   4982656315392,   2387522817792,   653726485824,    78310985281;
  9: 2641807540224, 22785590034432, 87344761798656, 195312592355328, 280761851510784, 269063441031168, 171901642881024, 70602460468992, 16915172820696, 1801152661463;
%F Formula
The row polynomials satisfy P(n,x) = (24+23*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[(24 + 23*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] = (24 + 23*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.