Real-rooted atlas

A038226 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 14\,x + 11\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 9 + 66\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0966121196
127297108913312744
28111886534159721464138416
3243445532670119790219615161051537824
4729160381470157187401976535289891817715617529536
5218756133617463377338513835745304386393720278119487171105413504
66561192456246985218112248830144702435091124464333724676921042143588811475789056
7196836495399526572815051164482781381643686506401790034863138434045787689787235794769120661046784
859049216513035724645349307640224139069098621190363013425261063138434040868153468057073843073025937424601289254654976
917714771449291309903651440894015105665561105424165469819888606722652089208083095496881485511671841070458559350118332853116706114049565169664
1053144123383404471565314576357606047549502495278957081304119331640335637504229819768594719333695140062092845401540683021299410271220141996313842837672156693912375296
1115943237599606316719133862247794663420604784414513599157713576648477104412243777493828446703881080282113656054052426520028879276892220028879276892212239870669211934522712143931793714773254144
1247829692455257425851696851858248871488654009454096346273599666349045047981631462664962970644692716756197471147108540403826210303232407368128040430987649082570372840534499144995391004510237974983358324111112006825558016
1314348907789189885202558737153218433268053540276594855285582311984971745225239908158227490416709853016746486126945860331405302869518927290916663129315454848611052153855559260801748532623962976014795170887425112458454177248169415651155568095557812224
144304672125254076326944870988011883445912801416110637942012461773613889683770811515591243879948889119201810047891676167058994153506482480151418327333305032302836654666610064462667111296209820521983407616236720410129820269900280200507912131951248459497298635721612177953337809371136
Technical details
Node id
n-45494a7331411e11 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4f0d71d8efeb5cf0

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:      9,      66,       121;
  1:     27,     297,      1089,       1331;
  2:     81,    1188,      6534,      15972,       14641;
  3:    243,    4455,     32670,     119790,      219615,      161051;
  4:    729,   16038,    147015,     718740,     1976535,     2898918,      1771561;
  5:   2187,   56133,    617463,    3773385,    13835745,    30438639,     37202781,     19487171;
  6:   6561,  192456,   2469852,   18112248,    83014470,   243509112,    446433372,    467692104,    214358881;
  7:  19683,  649539,   9526572,   81505116,   448278138,  1643686506,   4017900348,   6313843404,   5787689787,    2357947691;
  8:  59049, 2165130,  35724645,  349307640,  2241390690,  9862119036,  30134252610,  63138434040,  86815346805,   70738430730,  25937424601;
  9: 177147, 7144929, 130990365, 1440894015, 10566556110, 54241654698, 198886067226, 520892080830, 954968814855, 1167184107045, 855935011833, 285311670611;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3+14*x+11*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 66*x + 121*x^2.
%t Mathematica
P[0, x_] := 9 + 66*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3 + 14*x + 11*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] = 9 + 66*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((3 + 14*x + 11*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

Candidate equalities ?

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.