Real-rooted atlas

A013612 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 630\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04416302251296
192611984514175337546656
2194481555660595350283500506251679616
34084101145860752083725014883750531562575937560466176
48576612136756909065637337562511750033488437595681250113906252176782336
5180108854190054427051929737722522973068125164093343757032571875167442187517085937578364164096
6378228593612161306249205403265623007718950890006891920437503938240250001406514375002870437500025628906252821109907456
77942800465815106086013735145888171821002431469530350026051459253750186081851812508861040562500271256343750048438632812538443359375101559956668416
8166798809782011191420069871503829564510301257294408591050009118010738812507815437776125004652046295312501898794406250005086056445312580731054687505766503906253656158440062976
935027750054222127521803614031659829215576439875210626048066568753008943543808125030089435438081250214924538843437501096553769609375039162634628906259324436816406251332062402343758649755859375131621703842267136
107355827511386641630499500975997802476962325262848505897529345863925009478172162995593751083219675770925000902683063142437500552663099883125000246724598162109375783252692578125001678398626953125021797384765625001297463378906254738381338321616896
111544723777391194611434386364720394995614737013451597855016100255114208515250287504555610866343753696487143568281562535204639462555062500251461710446821875001347116305965117187553456996268457031251527342750527343750297534302050781250354207502441406251946195068359375170581728179578208256
12324391993252150868132439199325215086810150610568295641474475430315909416118498500845263393495947050625120751913356563864375012937705002488985468751056139183876651875000660086989922907421875314327138058527343750112259692163759765625291583616009765625005206850285888671875572181350097656250291929260253906256140942214464815497216
136812231858295168230172988198481733945322536494099240866972661251129579262217311058562524205269904656665540625380368527073176172781254528196750871144914062541585480365143167578125297039145465308339843751650217474807268554687570723606063168652343752296220976076904296875546719280018310546875901185626403808593759195771697998046875437893890380859375221073919720733357899776
141430568690241985328321163493564599084037522408758583817808073438700029195279392693578129000067774755733038663513750011618529554235199459500001521474108292704691125000155252460029867825625000012475644109542950273437507921043879074889062500003960521939537444531250001543060495923679687500004592441952153808593750010093279015722656250000154488964526367187500014713234716796875000065684083557128906257958661109946400884391936
Technical details
Node id
n-1d9325b8d0810008 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f96e0317bce1f626

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:             441,              630,              225;
  1:            9261,            19845,            14175,              3375;
  2:          194481,           555660,           595350,            283500,             50625;
  3:         4084101,         14586075,         20837250,          14883750,           5315625,            759375;
  4:        85766121,        367569090,        656373375,         625117500,         334884375,          95681250,          11390625;
  5:      1801088541,       9005442705,      19297377225,       22973068125,       16409334375,        7032571875,        1674421875,         170859375;
  6:     37822859361,     216130624920,     540326562300,      771895089000,      689192043750,      393824025000,      140651437500,       28704375000,       2562890625;
  7:    794280046581,    5106086013735,   14588817182100,    24314695303500,    26051459253750,    18608185181250,     8861040562500,     2712563437500,     484386328125,     38443359375;
  8:  16679880978201,  119142006987150,  382956451030125,   729440859105000,   911801073881250,   781543777612500,   465204629531250,   189879440625000,   50860564453125,   8073105468750,    576650390625;
  9: 350277500542221, 2752180361403165, 9829215576439875, 21062604806656875, 30089435438081250, 30089435438081250, 21492453884343750, 10965537696093750, 3916263462890625, 932443681640625, 133206240234375, 8649755859375;
%F Formula
The row polynomials satisfy P(n,x) = (21+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 630*x + 225*x^2.
%t Mathematica
P[0, x_] := 441 + 630*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 15*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] = 441 + 630*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 15*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.