Real-rooted atlas

A038281 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(23 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 1104\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
052911045762209
112167380883974413824103823
22798411168032182822412718083317764879681
36436343335809207008192073128960381542407962624229345007
4148035889926833392241782624033639321602632642560109884211219110297610779215329
53404825447248700293527785400492813539826944014128515072088456790016307675791364586471424506623120463
6783109852816537264858242387522817792498265631539264991169331205425349787648283061728051284391074201611007531417623811286661761
718011526614631691517282069670602460468992171901642881024269063441031168280761851510784195312592355328873447617986562278559003443226418075402241119130473102767
84142651121364943227663875112020298207384835205648196837519360103140985728614401291504516949606411230474060431360669643173789696026203428539596806076157342515206340338096537652599132235830049
99528097579139271093659896040333657060516315147840178624224986549760372780991276277760544584404647084032568261987457826816423549307421982720220982247350599680768633903828172801604105538424012815216811431690242472159215084012303
10219146244320203212744092102792109761574870250298080384547780956625419264012860944199031582720214721850975135989762614005142306003353623379921769693446144152477750671913779207071431915219189760221366564302513766441998399551465062436520347436056576116191483108948578241
115040363619364673836837362822790340152428078368035569122561637865060310003599364272691461678270259208025229180195707617281116553625070707146752116509943485639006617691181694901804439961652858953566263443456022062867575483872051262787607329440268288109195838833809162248764883384653578245460999706120583177327
121159283632453874980916935621761065304068811486769542287771455364794477721998374172672137580665066040302346242871248662247797614182444941283409095962656768535945740033939430440964893417626396838277939234041166096673657585664177606083982645170012166739203186693255462912175805300522432751206428222924498584521932821035720123168587776256666986187667409334369
132666352354643912456074173421076833949931240304841191699175473238401378412345074532574643204315029949798536755404809905807884754901768929281722749197348678568509440231126600389636379377664024117558301527274369843201957367050558735311175680122548197948025167308390458125627485229328367616020217609560079766388736048684544760058300334080725732344249316278272050485728295604610662412063348350820368238715343
1461326104156809986489611023879304183262383130888012968467521183867980803901967253749446057451520132327585127155127165747203313943001445276228150886463397170462431371321147392945050988259846529010892801109407681870254621012787201029015820865163706446643207516289474145543594914611242780461829128785678565376186002007952733850776371205971970823900484840980480133534751341874195202048018578748012782496723763212116574790945106558976566977372488557307219621121
Technical details
Node id
n-3e24a2a5ce552865 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-723ac5bf98d71a32

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             529,              1104,               576;
  1:           12167,             38088,             39744,              13824;
  2:          279841,           1168032,           1828224,            1271808,             331776;
  3:         6436343,          33580920,          70081920,           73128960,           38154240,            7962624;
  4:       148035889,         926833392,        2417826240,         3363932160,         2632642560,         1098842112,          191102976;
  5:      3404825447,       24870029352,       77854004928,       135398269440,       141285150720,        88456790016,        30767579136,         4586471424;
  6:     78310985281,      653726485824,     2387522817792,      4982656315392,      6499116933120,      5425349787648,      2830617280512,       843910742016,       110075314176;
  7:   1801152661463,    16915172820696,    70602460468992,    171901642881024,    269063441031168,    280761851510784,    195312592355328,     87344761798656,     22785590034432,     2641807540224;
  8:  41426511213649,   432276638751120,  2029820738483520,   5648196837519360,  10314098572861440,  12915045169496064,  11230474060431360,   6696431737896960,   2620342853959680,   607615734251520,    63403380965376;
  9: 952809757913927, 10936598960403336, 57060516315147840, 178624224986549760, 372780991276277760, 544584404647084032, 568261987457826816, 423549307421982720, 220982247350599680, 76863390382817280, 16041055384240128, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (23+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1104*x + 576*x^2.
%t Mathematica
P[0, x_] := 529 + 1104*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 24*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] = 529 + 1104*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 24*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.