Real-rooted atlas

A143491 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 8\,x + 3\,x^{2} - x^{3} + n + n x - n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 12 + 32\,x + 21\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
012322165
160240315136751
2360192037803264104510369
32520168004410057120365759276166391
420160161280529200913920877800445248932893040897
5181440169344066679201439424018433800140253125877207104737662320735
61814400193536008890560023030784036867600037400832023508828083790080129755611414919681
719958400239500800125737920038000793607299784800925670592077579132404147608960128458053917572114035239232359
82395008003193344000188606880006514421760014599569600022216094208023273739720016590435840077074832340210865368002581284541954978797761
931135104004566481920029967537600011644528896002982483504000529483578624066562895599205931080812800367390034154015076873812003691236893634086429218427969171962447
10435891456006974263296005034546316800217364539392006263215358400012707605886976018637610767776019928431531008015430381434468084430493347200310063899064926865201086912693347907421880179897765505
11653837184000113331778560008924877561600042386085181440013570299943200003097478934950400519190585673760064767402475776006017848759442520411598655067600020154153439219806693571059739201352028419470951254854032087629614610818338391
1210461394944000195279372288000166597714483200086313482551296003039747187276800077092809047654400145373363988652800207255687922483200224666353685854080184396197470284800112863259259630880499786639127193601514271829807464028108730318762242412533676791851060721718024871169
131778437140480003556874280960000326787824563200001834161504215040007046686661414400001965866630715187200411891197967849600066062750525291520008184274312841827200783683839248710400057560262222411748803186139824435859200128713105533634440035838631156421856061519608758192175490923473385769640290776942335685311
14320118685286400068291986194432000672249239101440000406337317856870400016912047987394560000514699626950885376001186246650147406848002114008016809328640002946338752623057792003224299224337551360002762892586675763942401835216538875054899209267343598421679680034405085910164981760885882366117967320014138596033510164481053943721929694891617545902751122323457
Technical details
Node id
n-6500fa0cff3eab18 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-21668cd305a13e34

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         12,          32,           21;
  1:         60,         240,          315,           136;
  2:        360,        1920,         3780,          3264,          1045;
  3:       2520,       16800,        44100,         57120,         36575,          9276;
  4:      20160,      161280,       529200,        913920,        877800,        445248,         93289;
  5:     181440,     1693440,      6667920,      14394240,      18433800,      14025312,       5877207,       1047376;
  6:    1814400,    19353600,     88905600,     230307840,     368676000,     374008320,     235088280,      83790080,      12975561;
  7:   19958400,   239500800,   1257379200,    3800079360,    7299784800,    9256705920,    7757913240,    4147608960,    1284580539,     175721140;
  8:  239500800,  3193344000,  18860688000,   65144217600,  145995696000,  222160942080,  232737397200,  165904358400,   77074832340,   21086536800,   2581284541;
  9: 3113510400, 45664819200, 299675376000, 1164452889600, 2982483504000, 5294835786240, 6656289559920, 5931080812800, 3673900341540, 1507687381200, 369123689363, 40864292184;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+4+(n+8)*x-(n-3)*x^2-(n+1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12 + 32*x + 21*x^2.
%t Mathematica
P[0, x_] := 12 + 32*x + 21*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 8)*x - (n - 3)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], 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] = 12 + 32*x + 21*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 8)*x - (n - 3)*x^2 - (n + 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.