Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 48\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1164836100
2256153634563456129610000
3409636864138240276480311040186624466561000000
46553678643241287681238630423224320278691842090188889579521679616100000000
5104857615728640106168320424673280111476736020065812482508226560214990848012093235204031078406046617610000000000
61677721630198988824914165761245708288042042654720100902371328176579149824227030335488212840939520141893959680638522818561741425868821767823361000000000000
7268435456563714457654962159616329772957696136031345049640809403514889182115790848157407699271682065976052940820659760529408154948203970568451720216576316939508121673139886489678364164096100000000000000
8429496729610307921510411596411699208117488189440395727549235201424619177246723917702737428488395077294489601416669293445120188889239126016019833370108231681622730281582592101420642598912046809527353344015045919506432030091839012864282110990745610000000000000000
968719476736185542587187223656679866368189253438930944106455059398656044711124947435521453111560791654437365725848928256770668095634145281284446826056908801734003215176826881891639871101992961655184887214243841145897229609861126138735158624256024554940634497024690607705345228812187194800209921015599566684161000000000000000000
101099511627776329853488332804700412208742404230370987868160269686150476595201294493522287656964854350708578713601456305212573614080354974395564818432070994879112963686401171415505363900825615973847800416829440179705787754689331201658822656197132288012441169921478492160746470195288709529634990790404158259201234969073087938560308742268271984640487487792008396803656158440062976100000000000000000000
111759218604441658054213946572891435386965852169143538696585216065147713213169664035179765135111618561495140018242243788851261943482591215616144174216044787793920336406504104504852480655992683003784462336107344257218801093836814759835367585150402561703057927029055815680164223442963516096512013137875437081287720968621730755584595066884564445694133020917761901852372555425382406005849597543448576013513161594472759296193045165635325132813162170384226713610000000000000000000000
1228147497671065610133099161583616174795960537317376192275556591049113615141700081545117696908502004892707061764315384523240358543361664505458964138295296530561115044819081625614148296401195175510016318336669026891448975366077336408695200389529698756716641297006329856136740069195642008764416161157938694863796043776161157938694863796043776135977010773791327911936959837723109115255848965599053384803172325785626521831822751868911616994568693353195084185628416248381519859548165812414441674516725767581410141314587033647383813383216168961000000000000000000000000
1345035996273704961756403854674493443293257227514675200395190867301761024003408521230477688832002249624012115274629120118105260636051918028805061654027259367915520018032142472111498199040054096427416334494597120013794588991165296122265603009728507163337335767040564324095093125750456320091160046130428005842944001269729213959532938526720015236750567514395262320640157128990227492201142681601386432266713166480670720010398242000348748605030400656731073706236754001920034478381369577429585100801477644915839032696504320503742584945124782899200131411109116119508582400246395829592724078592002956749955112688943104170581728179578208256100000000000000000000000000
147205759403792793630264189495929733126128498372925770956879670478848035022438474691073920032833536005377757322242364014592030922104602893593083904014577563598506979596697605739915666912123216199680191330522230404107206656005452919883566517055389696013384439714208723681411072028441934392693537822998528052508186571126531365535744084388156989310496837468160011814341978503469557245542401439872928630110352289300480152457133619658743183572992013975237248468718125160857601103308203826477746723225600744733037582872479038177280425561735761641416593244160203109010249874312464957440794774387934290787906355202483669962294658712207360059608079095071809092976641031678292030089003532288114630921336676555948032614094221446481549721610000000000000000000000000000
Technical details
Node id
n-c1a4b2af6b8ff4f9 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-6b852c5cd7d4c499

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:          16,            48,             36;
  2:         256,          1536,           3456,            3456,             1296;
  3:        4096,         36864,         138240,          276480,           311040,           186624,             46656;
  4:       65536,        786432,        4128768,        12386304,         23224320,         27869184,          20901888,           8957952,           1679616;
  5:     1048576,      15728640,      106168320,       424673280,       1114767360,       2006581248,        2508226560,        2149908480,        1209323520,          403107840,           60466176;
  6:    16777216,     301989888,     2491416576,     12457082880,      42042654720,     100902371328,      176579149824,      227030335488,      212840939520,       141893959680,        63852281856,        17414258688,         2176782336;
  7:   268435456,    5637144576,    54962159616,    329772957696,    1360313450496,    4080940351488,     9182115790848,    15740769927168,    20659760529408,     20659760529408,     15494820397056,      8451720216576,      3169395081216,       731398864896,       78364164096;
  8:  4294967296,  103079215104,  1159641169920,   8117488189440,   39572754923520,  142461917724672,   391770273742848,   839507729448960,  1416669293445120,   1888892391260160,   1983337010823168,   1622730281582592,   1014206425989120,    468095273533440,   150459195064320,    30091839012864,    2821109907456;
  9: 68719476736, 1855425871872, 23656679866368, 189253438930944, 1064550593986560, 4471112494743552, 14531115607916544, 37365725848928256, 77066809563414528, 128444682605690880, 173400321517682688, 189163987110199296, 165518488721424384, 114589722960986112, 61387351586242560, 24554940634497024, 6906077053452288, 1218719480020992, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (16+48*x+36*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[(16 + 48*x + 36*x^2)*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] = (16 + 48*x + 36*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.