Real-rooted atlas

A038337 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(408 + 506\,x + 121\,x^{2} + 132\,n + 121\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 408 + 506\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04085061211035
16480134648349133129624
29676828512029620812245214641815189
31389312532224078408005430480168371516105121827598
4194088969169459217563392017249760089602920222250381771561572834527
5265752576149448499235302417924507937280332057880013798849682852213211948717114803588900
6358318080023386226688657573396481035537592329917462016058442186880202383128643585639464214358881377935624617
74772796825635473489920011576182210562169992208384256295554099219636574791689642960835202862275676484437228836723579476919553940204282
86294932029445250076508160195104194560004244600143872059674785730560563850219018243600038711808015153224169600393562905516054232796893025937424601239553303974579
98235009441792761686775562243176296287436807869202513920001283991543521280144412981467955211370979416867846222924058982402291925155652005291234618604065621684240532853116706115965417614765456
10106993205379072108702124631654450271327187107841397570366472192025968368295936000338973767489617923177085592295014421442418328950784102678246973209603361490228289600698442969655728787460210886363138428376721147685512476658685
1113819955694796801530002836920729677722019111632896239626659591880704499631406013808640742695333263769600807887479183589376649033199568838656383283227629995264163144325746321920480693102645412809079758605524464938390084639579345227121439313637827655715373286
121776087209292595221282731769987072011781021844289617923989730314397155328922562639428740710415388647305225306112190625135537700864001777352454203896627212493889091700144128655840189500214118425124226164933575686729703437035779201165235687708972881111631331034578237974983358324189214436062754726791
1322725356822514892829305438953327820801755825371023933440064795620143592898560164576375468882657280304445671011484434432423187800893695918080449330676624580608000366578943679553679360229054633347835975680108213631267535329536376863392474003635209253342225924196400147895298824600404013101369258621814541772481694156512179453229013284964092
1428965200560222371843999662800762621132825788786278928482304010300842176673742848002851007286318087536640579308841650466953625689304063496702100766721064015042247006879744098852748857407733760007168654898622383063040403136154692191317196817314181002805652725765527329756285386649601252760178278968128001859255185223547936015372273263449595684594972986357216153060411857413857875793
Technical details
Node id
n-22ddc8a781b8d19b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7850c4e2593e77fa

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:           408,            506,             121;
  1:          6480,          13464,            8349,            1331;
  2:         96768,         285120,          296208,          122452,            14641;
  3:       1389312,        5322240,         7840800,         5430480,          1683715,           161051;
  4:      19408896,       91694592,       175633920,       172497600,         89602920,         22225038,          1771561;
  5:     265752576,     1494484992,      3530241792,      4507937280,       3320578800,       1379884968,        285221321,        19487171;
  6:    3583180800,    23386226688,     65757339648,    103553759232,      99174620160,      58442186880,      20238312864,      3585639464,       214358881;
  7:   47727968256,   354734899200,   1157618221056,   2169992208384,    2562955540992,    1963657479168,     964296083520,    286227567648,     44372288367,     2357947691;
  8:  629493202944,  5250076508160,  19510419456000,  42446001438720,   59674785730560,   56385021901824,   36000387118080,  15153224169600,   3935629055160,   542327968930,   25937424601;
  9: 8235009441792, 76168677556224, 317629628743680, 786920251392000, 1283991543521280, 1444129814679552, 1137097941686784, 622292405898240, 229192515565200, 52912346186040, 6562168424053, 285311670611;
%F Formula
The row polynomials satisfy (11*n+23+11*x)*P(n,x) = (132*n+408+(121*n+506)*x+121*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 408 + 506*x + 121*x^2.
%t Mathematica
P[0, x_] := 408 + 506*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((132*n + 408 + (121*n + 506)*x + 121*x^2)*P[n - 1, x])/(11*n + 23 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 408 + 506*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 408 + (121*n + 506)*x + 121*x^2)*v[i - 1])/(11*n + 23 + 11*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

Proof chains ? JSON

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.