Real-rooted atlas

A034870 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(36 + 64\,x + 16\,x^{2}\right)\,P_{n-1}(x) + \left(64\,x + 32\,x^{2}\right)\,P_{n-1}'(x) + \left(16\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1366416116
2129687049856307225623184
346656953344278809623961607731209830440967059776
41679616983203846387343361148485632802185216253231104380764162621440655362983399680
56046617699395338241346575605764554404659205822350950403435795578881038275051521682964480014588313606291456010485761648092623872
6217678233699782321766427348044578816164957911252992356826781515776351335966834688179722337124352515321836339208614730268672847081635840477060136961409286144167772161142234168922112
7783641640969992163583590454560425948610565696287812354048019953499676737536030865318221132595224615858848713932811120149805400064030156496788520960507548062973952053695434509516835395630792704139848161689630064771072268435456963872942447214592
82821109907456999717889009254410789780794596392961915594471034270515210577629561669392793624787167660416906035229387360240204041420819650398131851362304079510550217970679808203824507720512307203405268524506021888375887235574136832273938243261562881293987746938880378386618777606184752906244294967296967973361754826670080
9101559956668416999898440043331584212439477885296705536634479523518411177984054277941881406048174080188176797258189002244096321701918451037345153024306356191444790799237120176108358772914231705600645420829327452064972801565233967310430640537625734604146984741765122907337781260090081282263960384800659865612048671240072724484277127719839334496291448789401612369505812480687194767361136261947537682756534272
1036561584400629769999634384155993702441734129540721620811776208292190774401088828211227289090139876149948317696137730396511528672554909696332551776261870443911184384439514277189774748313190400347678283899404498285625344175108608163811751194787840586568976809816527481077761347682279814216240490086421687766642996764276162562475837265803218203443202015025952174326677504011654373125360304783364726372977072301670413067145940303872002334538063675392024189255811072010995116277761536568633364421826875228160
1113162170384226713699998683782961577328648189489097488762474070016680210266631773360226304000135418548790117498630438912009837858009917350761140846592033058483289650384057153880064059541544549860290145218985984063394838220158515014122078208042718086408380976453201690624019131907810900938560813137920059018043435473348686642151424128690214000207024673374863362020112766592732241688788992230998212988930414090387456193549174338246504779612161187994326244035630465024530264744654233455820801692735445296054534144374749626727451852805442098772374978564644337115725824175921860444162365216221007495089596802269184
124738381338321616896999995261618661678383104160609985257806113753635225622143654099106718304852718387266632224359283304915490371010566911483964807765549278017093632031977982122713251021714911920128077450402949947671338682794639360010921904695290306538712981097676809659371736763253191029035106304005653420075513902594643354818969602278465284078839320226400986726406513867772136143161812934721536013505269628994799545394554470400206351560109429122725146787840023490290965755626536329805824020046218690419271074735718400128433638503041717120270336061546608144063159987077120218375592544778705698816056291918292870171197440102034228697650062950412281034708862631936878201927337246722814749767106564102564895819961246865224433139712
1317058172817957820825699999829418271820421791744314891570726229475891395690496719510360621599450319143147929603260455203145523956175698887966720480045132944545916424255871112970243033901747445635073136817501645045769782961297699102457786477632815104001805296675635403677809035304973107200206580618908631929955052643501998080015537486206252727918126966851921510408021626466982648947965515279001190402937740121739918944713128252381593607825939910563456994799147036966912015456902236065742165234098225807360229584137629250899300159076696064025902990405427369625986264989696022338094564628964814366546329600147606558509368224264839233536074622786215398017302160998400286856473989999124826554368082834377809854848533790720176015783858813741826048026600736379045203148800269681175186479513600163931026436286054445035996273704967950106747617128493762925052481765376
146140942214464815497216999999385905778553518450278461728347848694439020347298676736233524434051900502556034633899704321589577522357510805665758814473814016330782096026322391089859417027630858242838563355313153134935157178431921192961209346282961335775458698151789319946240289345192778717007765751644420887386521642374297918279149852384782931791446016004043780606611628079762396055873194557440263509898609114809784026276982440329216012152046584558429251827167809639966310404077099783924753003449362764766642176001016812279609251767697367845612552192001916676936568598228128265119840862208027654946743461178785099379333346099203082580925918500042571775463129088002670642297987275251156548153507840018038551020063563406645663891456009497946651454671878159842934784038843926165957863887372235571201224158925682178107208682700802932624737177717244805775360522488774672864940707020806684058099502553067683845785756426884366729216302641894959297331207205759403792793617084520128326464921227776204798989172736
Technical details
Node id
n-2137bba207333614 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-14e66869ea533ba5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1;
  1:              36,                 64,                    16;
  2:            1296,               8704,                  9856,                   3072,                     256;
  3:           46656,             953344,               2788096,                2396160,                  773120,                    98304,                     4096;
  4:         1679616,           98320384,             638734336,             1148485632,               802185216,                253231104,                 38076416,                  2621440,                    65536;
  5:        60466176,         9939533824,          134657560576,           455440465920,            582235095040,             343579557888,             103827505152,              16829644800,               1458831360,                62914560,                 1048576;
  6:      2176782336,       997823217664,        27348044578816,        164957911252992,         356826781515776,          351335966834688,          179722337124352,           51532183633920,            8614730268672,            847081635840,             47706013696,             1409286144,              16777216;
  7:     78364164096,     99921635835904,      5456042594861056,      56962878123540480,      199534996767375360,       308653182211325952,       246158588487139328,       111201498054000640,        30156496788520960,        5075480629739520,         536954345095168,         35395630792704,         1398481616896,          30064771072,           268435456;
  8:   2821109907456,   9997178890092544,   1078978079459639296,   19155944710342705152,   105776295616693927936,    247871676604169060352,    293873602402040414208,    196503981318513623040,     79510550217970679808,    20382450772051230720,     3405268524506021888,     375887235574136832,     27393824326156288,     1293987746938880,      37838661877760,      618475290624,      4294967296;
  9: 101559956668416, 999898440043331584, 212439477885296705536, 6344795235184111779840, 54277941881406048174080, 188176797258189002244096, 321701918451037345153024, 306356191444790799237120, 176108358772914231705600, 64542082932745206497280, 15652339673104306405376, 2573460414698474176512, 290733778126009008128, 22639603848006598656, 1204867124007272448, 42771277198393344, 962914487894016, 12369505812480, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (36+64*x+16*x^2)*P(n-1,x) + (64*x+32*x^2)*d/dx P(n-1,x) + 16*x^2*d^2/dx^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[(36 + 64*x + 16*x^2)*P[n - 1, x] + (64*x + 32*x^2)*D[P[n - 1, x], x] + 16*x^2*D[P[n - 1, x], {x, 2}]];
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] = (36 + 64*x + 16*x^2)*v[i - 1] + (64*x + 32*x^2)*deriv(v[i - 1], x) + 16*x^2*deriv(deriv(v[i - 1], x), 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.