Real-rooted atlas

A038301 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(64 + 44\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 4096 + 5632\,x + 1936\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
040965632193611664
1262144540672371712851841259712
2167772164613734447579136218071043748096136048896
31073741824369098752050751078403489136640119939072016491622414693280768
4687194767362834678415364872103526404466094899202302830182406332783001672563138561586874322944
5439804651110421165598834688436540475965445002026287104034388930723840141854339235843250828607488319277809664171382426877952
6281474976710656154811237190860837251453949050885122074917994496440178313265152024209807229583368322121235169281634702385479681404822362521618509302102818816
7180143985094819841114640907774197763065262496379043844917191921274716165070854168814551043486212241060003841597847277152501764707942870181478480917768081244166181218395095041999004627104432128
811529215046068469767926335344172072960245220999710323507204495718328022597632054089111134021877760446235166855680491522556555643444002816010043611456387153920258936857859981312039559797728608256027197360938418176215892499727278669824
97378697629483820646455801400822971393638419181731532896416563203956232128659885916160543981917690734313472052358259577733177671683599630345969155964928176767561632413908992060763849311142281216013925048800470106112019146942100646395904119668388129039974423316389970546096340992
10472236648286964521369638959523483674573012992147315698172644479205376337598474978976931512320522222640983104940933120574444905081415435026432460752684284051963510784271514974667387764211712116666590677393179934720356481249292034716467207352425766648216027136919053220831027003392526540907767775887362518170116818978404827136
1130223145490365729367654427011936282014370622341121114242371633092788171571228088193118250880701825024482765819219937012062617605974227012846720524274892854763747617761604805853184376500764872111033040240641941332068871822514113740874148099852743221025177602039072745950438578192384382326139865707233411072438082035262789538283522316779994178213904384271962372616449667721330688
1219342813113834066795298816186174576220652892904751104831967637486042615168106496228791100308661719171229286443255817402106356280810536965947674892789623988611448832613353973318929973825555660848192097903630212229150801922899055889515254954409852928132873394936115852077118259245675229509289824151509401611418807377322456037877350419626075179772971315101696207583487478367965832806410193831974384141179289629371936242576564113903714304
1312379400392853802748991242241276625665513048408489721856061437610152815454658567864320183032880246929375336983429120377505315509291836632528322560570976789707803902906699087872654244238206858638747259371520578305174843562546749809623040397584807704949250890494115840212597431897785363323389214720876964406578364623708980510722740513770557389449090564096062803440575273508208325427209964007398961662359974707209786078695408775532118016044852860687290221188874243172169114198268924301601144832
14792281625142643375935439503368715097876569077135289834536964493722342605930397883820933120144173591825273600265439254937603221378692345957005930908352512053152748423708290597859987816448669946099923823246077193596436486579827767108978309686722822144050890855386233504113983246827520310999671804760302918786508390401496685920560408957796660071628856125722021015335917374752686081607768078727001810133130936320340104785884558075220470005760501047229204929307244442419204592932934378518649740722176197352587024076973231046656342594264333413043824572923641856
Technical details
Node id
n-adb898de6c27baa4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-89a932e93fe55d02

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:                 4096,                  5632,                   1936;
  1:               262144,                540672,                 371712,                  85184;
  2:             16777216,              46137344,               47579136,               21807104,                3748096;
  3:           1073741824,            3690987520,             5075107840,             3489136640,             1199390720,              164916224;
  4:          68719476736,          283467841536,           487210352640,           446609489920,           230283018240,            63327830016,             7256313856;
  5:        4398046511104,        21165598834688,         43654047596544,         50020262871040,         34388930723840,         14185433923584,          3250828607488,           319277809664;
  6:      281474976710656,      1548112371908608,       3725145394905088,       5122074917994496,       4401783132651520,       2420980722958336,        832212123516928,        163470238547968,        14048223625216;
  7:    18014398509481984,    111464090777419776,     306526249637904384,     491719192127471616,     507085416881455104,     348621224106000384,     159784727715250176,      47079428701814784,      8091776808124416,       618121839509504;
  8:  1152921504606846976,   7926335344172072960,   24522099971032350720,   44957183280225976320,   54089111134021877760,   44623516685568049152,   25565556434440028160,   10043611456387153920,   2589368578599813120,    395597977286082560,    27197360938418176;
  9: 73786976294838206464, 558014008229713936384, 1918173153289641656320, 3956232128659885916160, 5439819176907343134720, 5235825957773317767168, 3599630345969155964928, 1767675616324139089920, 607638493111422812160, 139250488004701061120, 19146942100646395904, 1196683881290399744;
%F Formula
The row polynomials satisfy P(n,x) = (64+44*x)*P(n-1,x) for n >= 1, with P(0,x) = 4096 + 5632*x + 1936*x^2.
%t Mathematica
P[0, x_] := 4096 + 5632*x + 1936*x^2;
P[n_, x_] := P[n, x] = Expand[(64 + 44*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] = 4096 + 5632*x + 1936*x^2; for(n = 1, N, my(i = n + 1); v[i] = (64 + 44*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.