Real-rooted atlas

A038333 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 31\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 456\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144456361961
11728820812996685929791
220736131328311904329232130321923521
32488321969920623808098769607819260247609928629151
429859842836684811228544023704704028149336017827912847045881887503681
5358318083971358721886395392497798784078818140807487723376395185400489387173927512614111
6429981696544643481630182326272955773665281891635379202396071480321896889921928581168694416983563041852891037441
751597803527352687001646567017676817203925975044085932419072646939299686468288037189124633831094976183422480842832268769777926439622160671
86191736422498035826688069850526515202949244452864081718648381440155265431924736204864111567360185353243799040110053488505680387225237334806131066257801819628286980801
97430083706881294072912281610244743888896048662533472256015409802266214403415839502344192540841254537830461166570453683204842353494249920255568656640968080930074602973211649025889821925408476896404831
10891610044825616940590851686414752431200010247786005355560960277376440791859207026869833393766412980190108907929617615972290660761617432472579299712012267295518766464058269653714140704167745972813435362213314919066161787662783788549761
1110699320537907222022768107192322092162970183270412146168354675097648078583070588928013702396175117844482892728081413767168458015279557179801654389314447415101444784245252318920960303002199313531660813084185879447958083452771273743211164205298346225705924417546297445042591
1212839184645488642846019263083315229290281582565785618550511668958330888077201955858939904255778061935533096966074728970968911052810992366709372315238415229008045276228403216075064047791574425612726092371168329753673271440924908565248290032786994429737447064901221659185912799006685782884121756943935220796320321
131540702157458636836591676239642624040555774498937241602782576750343749632013217239564132810752046040051148395957452812149457941937822105602473282509608770928640391603064021388730368048225192143374723276804581393253620598711296329721484162088543616017401967219665784246406358411099493267320801438212034409191417801518112702987479829923465261991844685929951
14184884258895036416468373455867425587255619347884256788480410965181589230714880211475833026124972032080360816549927489372162332695924852061844275252763360204987113144320939847353651332952883201322748127361135267020801466045841158591587614721266130499182420007485448352944265439576438272040693831036756910853120138068355303282376108802914776389735961273408288441413567621167681727423121747185263828481
Technical details
Node id
n-efd41e40cbfa2719 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b68b92f4c09b2b20

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:          144,            456,             361;
  1:         1728,           8208,           12996,            6859;
  2:        20736,         131328,          311904,          329232,           130321;
  3:       248832,        1969920,         6238080,         9876960,          7819260,          2476099;
  4:      2985984,       28366848,       112285440,       237047040,        281493360,        178279128,         47045881;
  5:     35831808,      397135872,      1886395392,      4977987840,       7881814080,       7487723376,       3951854004,        893871739;
  6:    429981696,     5446434816,     30182326272,     95577366528,     189163537920,     239607148032,     189688992192,      85811686944,      16983563041;
  7:   5159780352,    73526870016,    465670176768,   1720392597504,    4085932419072,    6469392996864,    6828803718912,    4633831094976,    1834224808428,     322687697779;
  8:  61917364224,   980358266880,   6985052651520,  29492444528640,   81718648381440,  155265431924736,  204864111567360,  185353243799040,  110053488505680,   38722523733480,   6131066257801;
  9: 743008370688, 12940729122816, 102447438888960, 486625334722560, 1540980226621440, 3415839502344192, 5408412545378304, 6116657045368320, 4842353494249920, 2555686566409680, 809300746029732, 116490258898219;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+31*x+19*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 456*x + 361*x^2.
%t Mathematica
P[0, x_] := 144 + 456*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 31*x + 19*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 456*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 31*x + 19*x^2)*v[i - 1])/(1 + 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

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.