Real-rooted atlas

A316649 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 10\,x - 2\,x^{2} + 12\,n + 15\,n x + 3\,n x^{2} - 4\,n^{2} - 5\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 4\,x^{2} + 15\,n x + 6\,n x^{2} - 5\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 3\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(12 + 18\,x + 4\,x^{2} - 30\,n - 45\,n x - 10\,n x^{2} + 12\,n^{2} + 18\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(18\,x + 8\,x^{2} - 45\,n x - 20\,n x^{2} + 18\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{2} - 10\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 7\,x + 2\,x^{2}\)
\(P_{2}(x) = 26 + 196\,x + 229\,x^{2} + 72\,x^{3} + 6\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
137212
226196229726529
32525238149641252539424902037431
425681400468494861446813967041294220428602870703745974
5269283784122458793421433278591723964319684838328087334439481036936215498252495130321
628764810344662224332809781323169985326503333389245608811961195366993932860925585296756085043780027603667946492482655646345
7311289628572279541283772460621179569964895376691526703154243356698714082246816362175673764810945697336300474162294136759675705648647722218722748392964343216880280963742
83401331279596165718677476222705010332025907328951211840710186511051587132462241226564760483055776462189904247299867423103421738211888866361190058912247212723914478669009680941841457015601183465141891890128704117434677537407
937441612822329312629223584207124430628972825636866539678358696001546912142573562098459073401862463205317703048354929526614541664449372380887115834894473230613513608912548141990216114483818400223664573462290217912626417686928891953132321961840207089754108921770486201177267904262026851
10414589593663000061506846190250786644208347763824130630618378848374185236726461402465751592593817328953979011332530119111068316822191210915482767895753729040350397193818893390636241275449898072155238528226949527289298064392425967746620764864153299538218438504909507696062170963110424688599182853404232595065560407140476641385344184756388669810305109740320
114612784025617858927649562741013312892970453902671091357116114383351143783528192689398311740684683109958138267122751957355779640565610380963235652271536211093816160285520017661402024407451712689926754422104455874598740015162803095440771866252225463651170249900292695016989578580132581393241098676447635462467289599067876521374653639773556284344233943425823478873086431563288158569377395022594701428189408492705432146339185231444292111133
12515268544512508254431882827745414618518635799511458032378011963064914971470938010844239824534491345027183726590084317080564321742927791633680267912629550654778167845983787166764669553683512372781611394520082749262781038590677745477393663905999082143002012318785363957314317826098985813659635199409165488023893397025261532245080378350362398235342711301596218059056784225367675581280645556140913543319792826446601271623910660842293090801476360657946311244107010308465823656121900060464857217452270415662188020836508361939103437
1357750881034241451290611134066346290185706417339216520650241653818350530346146718865476453410696898508083242125089928392622572533107137641009423142869382958026640760905449922422659995657707009458488957383699445936731755483672623632885165310034153954853314836993762608467962661748230206590983807423619936182254077741667993336367240495614754735564144868327459611905519932093960359426624409385758405073646294727017025148826888679788611132259811127059639923943433168258685148476194987737863983067146734130875192040469020788845748288417843367336560296846440619496766064223933813238430217001040060029565581047331885159603084132
146491216488857641559106465015823166155931879442189728855234435625526146657976433038372416688246542563273222181581433414968887618773046331940603318008597879174685817299547671290230188352467763767280611346286285645571477399102708720922239337459598358867361282502558783553275264893673718495714974499220669468286729349928369894610597512388105591602122208540437131487895505739181395257277146075857384955941497661955780199138546949949248234748783821141941945505563528878140325424210746359017729038138256804178106079956160409447444961175472570414737578218950605663508294833069844622427088410059612658216750359575207522384261765278458749058708309150419347238563890823481636003315954208300170896716004011660015606291972192964125212251905857
Technical details
Node id
n-80eea50c2afaf945 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-0ea4d4f1315e31da

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:         3,             7,               2;
  2:        26,           196,             229,               72,                 6;
  3:       252,          5238,           14964,            12525,              3942,                490,                 20;
  4:      2568,        140046,          849486,          1446813,            967041,             294220,              42860,               2870,                 70;
  5:     26928,       3784122,        45879342,        143327859,         172396431,           96848383,           28087334,            4394810,             369362,             15498,               252;
  6:    287648,     103446622,      2433280978,      13231699853,       26503333389,        24560881196,        11953669939,         3286092558,          529675608,          50437800,           2760366,            79464,             924;
  7:   3112896,    2857227954,    128377246062,    1179569964895,     3766915267031,      5424335669871,      4082246816362,      1756737648109,       456973363004,       74162294136,        7596757056,        486477222,        18722748,         392964,         3432;
  8:  34013312,   79596165718,   6774762227050,  103320259073289,   512118407101865,   1105158713246224,   1226564760483055,    776462189904247,    299867423103421,    73821188886636,    11900589122472,    1272391447866,     90096809418,     4145701560,    118346514,     1891890,     12870;
  9: 374416128, 2232931262922, 358420712443062, 8972825636866539, 67835869600154691, 214257356209845907, 340186246320531770, 304835492952661454, 166444937238088711, 58348944732306135, 13608912548141990, 2161144838184002, 236645734622902, 17912626417686, 928891953132, 32196184020, 708975410, 8921770, 48620;
%F Formula
The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(5*n^2-15*n+10)*x+(n^2-3*n+2)*x^2)*P(n-2,x) - ((5*n^2-15*n+10)*x+(2*n^2-6*n+4)*x^2)*d/dx P(n-2,x) - (n^2-3*n+2)*x^2*d^2/dx^2 P(n-2,x) + (12*n^2-30*n+12+(18*n^2-45*n+18)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) + ((18*n^2-45*n+18)*x+(8*n^2-20*n+8)*x^2)*d/dx P(n-1,x) + (4*n^2-10*n+4)*x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 7*x + 2*x^2, P(2,x) = 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 7*x + 2*x^2;
P[2, x_] := 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (5*n^2 - 15*n + 10)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] - ((5*n^2 - 15*n + 10)*x + (2*n^2 - 6*n + 4)*x^2)*D[P[n - 2, x], x] - (n^2 - 3*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (12*n^2 - 30*n + 12 + (18*n^2 - 45*n + 18)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] + ((18*n^2 - 45*n + 18)*x + (8*n^2 - 20*n + 8)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 10*n + 4)*x^2*D[P[n - 1, x], {x, 2}])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 7*x + 2*x^2; v[3] = 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (5*n^2 - 15*n + 10)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] - ((5*n^2 - 15*n + 10)*x + (2*n^2 - 6*n + 4)*x^2)*deriv(v[i - 2], x) - (n^2 - 3*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n^2 - 30*n + 12 + (18*n^2 - 45*n + 18)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] + ((18*n^2 - 45*n + 18)*x + (8*n^2 - 20*n + 8)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 10*n + 4)*x^2*deriv(deriv(v[i - 1], x), x))/(n^2 - 2*n)); 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

References

  1. [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
  2. [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