Real-rooted atlas

A038320 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(17 + 58\,x + 41\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1394\,x + 1681\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0289139416813364
14913355478573168921195112
28352180573229148544686628282576111316496
314198571712180582587530199181690240189685115856201656356768
424137569349284822210598201567721774601224967393511817332502475010424138068692544
54103386736927482303501223719572014722794354859037327557031312838695652624046791947542738812207984167552
6697575744113459108474411361070976925480046000632165207269136703187528486872838437843518172264865812478167984925229121128063081718016
711858787649725740544957292483205513526813974117301611650553424355830212192296462288461960330019426772202622346545792412216935600555133273819343939617427658739644928
820159939004494862102936377052768117162444533937142018199601432347023415189041453807971780764833140258256378101148193297092823601038439526047186055565492884697337013422659310152401430804206899405824
934271896307633909213249102499109640421215301357932806946754156538264127625520049012919770151205004783115944565878854094536780366390895033064729397123607930455203735847192010095251003729099849898755032903171624844124986644000165537792
10582622237229761168617729833554362236664592792147541798102907930942140975735254450761249537651901583511728216105942115239881039196219006389487485173464330119925330400445295353854037609056686460256023803681846896674112267122470114681964225634903003661860811449225352009601191936
11990457803290593731053765244346261344936625000642236943973807426527382129423959721248179804015510401337812445114919673344743924001958523188806673534612237055592414591300699603699682039195504355779003819269151886042020456272138831812405517032947672357022498653135638092712390192510310231501362932184055070416556869132288
12168377826559400929568522779088800783889124306251273769931859787425012288133452570241365706679335568927505759992910415009794995061317390582660648432742690017681605989014165787882610842800873875479306007335080581797211541122195002171481922604921984171017947182006990412593397231409330327744219220174538350973243778398379292271949155588021614875194084160298409672704
1328624230515098157931035523633340315713351748207545698062410185182704827815111228358551321923165956393005182457013968798192155827497472819340399273317538503885874232443136011908998263521084429510927346040546355395505311740924726381490455723194511074557805285097162740902394132852945638765504387646697930778582586152807225363974908858174574596719529347034674945510551555098314991537910888601282761256881297307761016832
1448661191875666868481187774952179043916020833965175173562355397880382274716659309647027120299635917616782414508022017343631937347876227993744766860588602342370473056722642124717033280436083640807168706033715297653785760701536820639907593222510934320259451484502046620506257735234131050902408272002427488163429835997545929294361585820254521766333725069807828088013153855991196715792589434804229867416776983117616994726375903091465305434643264116400152899115243850138976256
Technical details
Node id
n-5dd6a785fd91bd58 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-72ca7b974ae5086b

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:            289,            1394,              1681;
  1:           4913,           35547,             85731,             68921;
  2:          83521,          805732,           2914854,           4686628,            2825761;
  3:        1419857,        17121805,          82587530,         199181690,          240189685,           115856201;
  4:       24137569,       349284822,        2105982015,        6772177460,        12249673935,         11817332502,          4750104241;
  5:      410338673,      6927482303,       50122371957,      201472279435,       485903732755,        703131283869,        565262404679,        194754273881;
  6:     6975757441,    134591084744,     1136107097692,     5480046000632,     16520726913670,      31875284868728,      38437843518172,      26486581247816,       7984925229121;
  7:   118587876497,   2574054495729,    24832055135268,   139741173016116,    505534243558302,    1219229646228846,    1960330019426772,    2026223465457924,    1221693560055513,     327381934393961;
  8:  2015993900449,  48621029363770,   527681171624445,  3393714201819960,  14323470234151890,   41453807971780764,   83314025825637810,  114819329709282360,  103843952604718605,   55654928846973370,   13422659310152401;
  9: 34271896307633, 909213249102499, 10964042121530135, 79328069467541565, 382641276255200490, 1291977015120500478, 3115944565878854094, 5367803663908950330, 6472939712360793045, 5203735847192010095, 2510037290998498987, 550329031716248441;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (17+58*x+41*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1394*x + 1681*x^2.
%t Mathematica
P[0, x_] := 289 + 1394*x + 1681*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 58*x + 41*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] = 289 + 1394*x + 1681*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 58*x + 41*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

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.