Real-rooted atlas

A038320 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(35 + 18\,x + 18\,n\right)\,P_{n}(x) = \left(901 + 1260\,x + 324\,x^{2} + 306\,n + 324\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 901 + 1260\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
090112603242485
12051948654340205832109025
2437257147736817515448164801049764587625
38936747393531306648156052546320183708001889568187578125
4177482125965168676212506902023933361601418750640396809280340122247510628125
5345167236722362747750608056265888925289884075390089040357525161288332994880612220032295960765625
66606452635349704082084816101178380002918670076224321310435824021712345643528580603870721714216089601101996057611516734140625
7124866058193910702453269186402603064886888694636325200011820613808707210410458120697658623333237504198579689579523471287581440198359290368443619451953125
8233618116699092247589047490209632207942267402415618389321280391258634634000042554209711345923123137436209280150745714039296044680430155392069425751628800357046722662416945362314453125
94334386885965354625638710641982222511315701529806357257241896484011957311027140336015493841931506400014042889204744153688340173195633920373095642247257609829694634186240137462988225024064268410079232642741533369140625
1079853518396784899362275673685156049956898074933405616020814730510145603432918910624101360516555836372462515255777830953423040004333234383178195968238518467628211584089542954139341824021232140409842278426992732233277440115683138142617624233838500634765625
1114623818154467001118685723304847664261095386253821163252038966380498448056368937217661734843517601606606050172079436482014567761852603809281864573206157284480001267471057079622320646201480158333501184020953051268605986816451665532354826649652635827854891008020822964865671168909010316667236328125
122664331490851697053368520217492568427722354401136410805696769201244532097771316802454881971402227551184472357701514361132870467477454107227336332167252443942669373713408587340559939544611200035489189598229424977921562772999900042298368480015356335337152512948497617945135964161020325278417887232037481336758208102433944270003428955078125
134832443622254806662371936950252995820431049750229361496737742202118961022769725127084062108400591659956388400132563626455720287763936212560965681462509791680260270180127876868709760244769983065091362827520176202167981863383360000958208119152194474403843835897363391012914176010800345517545085931520196995659111682084864019677701798059253760067466406164774584321262561262322662353515625
14873039030710493816865139174376320938431874241035892083643139814206404776022018703686823251201525651936394202091500480357744387407961348797184063630540698745738126689288745365445180171831429120936972648460356727355136078326394580829236104806405074622437877665440768000250876307559847280571187292061536721384309940224023926919300407574986752040524821302974600314880377811874522737672192012143953109659425177646797557608385894775390625
Technical details
Node id
n-5cdd32670c5ced64 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9d616bbf9c4c319b

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:             901,             1260,               324;
  1:           20519,            48654,             34020,              5832;
  2:          437257,          1477368,           1751544,            816480,             104976;
  3:         8936747,         39353130,          66481560,          52546320,           18370800,            1889568;
  4:       177482125,        965168676,        2125069020,        2393336160,         1418750640,          396809280,           34012224;
  5:      3451672367,      22362747750,       60805626588,       89252898840,        75390089040,        35752516128,         8332994880,         612220032;
  6:     66064526353,     497040820848,     1610117838000,     2918670076224,      3213104358240,      2171234564352,       858060387072,      171421608960,       11019960576;
  7:   1248660581939,   10702453269186,    40260306488688,    86946363252000,    118206138087072,    104104581206976,     58623333237504,    19857968957952,     3471287581440,     198359290368;
  8:  23361811669909,  224758904749020,   963220794226740,  2415618389321280,   3912586346340000,   4255420971134592,   3123137436209280,  1507457140392960,   446804301553920,   69425751628800,    3570467226624;
  9: 433438688596535, 4625638710641982, 22251131570152980, 63572572418964840, 119573110271403360, 154938419315064000, 140428892047441536, 88340173195633920, 37309564224725760, 9829694634186240, 1374629882250240, 64268410079232;
%F Formula
The row polynomials satisfy (18*n+35+18*x)*P(n,x) = (306*n+901+(324*n+1260)*x+324*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 901 + 1260*x + 324*x^2.
%t Mathematica
P[0, x_] := 901 + 1260*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((306*n + 901 + (324*n + 1260)*x + 324*x^2)*P[n - 1, x])/(18*n + 35 + 18*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 901 + 1260*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((306*n + 901 + (324*n + 1260)*x + 324*x^2)*v[i - 1])/(18*n + 35 + 18*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.