Real-rooted atlas

A200536 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 45\,x + 117\,x^{2} + 123\,x^{3} + 45\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
16334584
2363961629297020257056
32163564244628939718346520047591125592704
412962851227410415040085151681112800601541835012028500410062549787136
5777621384026438401935252092877030305366193696577725108857925011153700006766031251845281254182119424
646656153964823269680212984640131486814057680025481843601706943260019110739613328758985289500073626721875365365687508303765625351298031616
72799361077753619252598421150914401596483756087582972708360129252798112750879259727009693959854926542214825673516584562566922815093754568731846875191816985937537366945312529509034655744
816796167390310415234117121952889523217425071216011475165905285769431306352225905518697766961942290752116942913902332032453051098230048410856162900055134014394375046342983802500027113455518750098648735625000168151253906252478758911082496
910077696498845952116548554241707715975681758511342656135141733298888032591737273637757279090203214228291579335744330590624228033106712186845018052123846948823930033887496391623000427596890515987504173029846341875030393675446062500155573540114062504994092241015625756680642578125208215748530929664
10604661763325639680868445452801431687882240167109992697601468006361775361007056234542240552432981855744024611543329141620899276815477675802709641148135467496744576116082568501384399312264216125233057664220392000031863888671063062503483647909291137500297419884464234375019110726308953125008694260583222656252497046120507812503405062891601562517490122876598091776
1136279705621949221888633534359040116056510732801514197002700801496692747668096116387533682513287298486327922768037528967454164604016003418609833637405700917629768983534170353071281461249174275688222326737650590019229680314210375158325331447256923125230928668969431331250276193073094245381250266379153576670406250202120901178384375000116187555987228515625475687285956738281251236037829651367187515322783012207031251469170321634239709184
1221767823361436676341764541856344064915282552729601320013733195520144992608932188161260372673947087368893392144727242245183982437035699440252709292733397473601039048225454389764243623575082955045744241075247721908030640321271768131221628430818058446462681809424238501066117328719020591750016402444429683267759375211044364371945299250002243192576434147181250019354176985760223750000132150422721798808593756872370332861601562500255767894039759765625060678220728339843750068952523554931640625123410307017276135571456
13130606940169338396221443209012519731270551583452979211144898358663680134674790042048256129409983183827174410147731851056580352661237657325759891523627172607134891557601690648810959451996224674019601107280229973623085001119744588736998680953505695604228287531731375083980844155274853791360256228451293601507132424671235188109070001147660082726274281896875156914795634921536757187518060782518115837589375001727414753558532900468750134826858133831574648437583681119687491307617187539730076325559318359375013553308029957363281250029580632605065673828125310286355997192382812510366465789451195388002304
14783641640966034040635392223945189945344533409192168652891595468325344256120757859864203161612710387488557088512109652051108894621184789852237913045315008481503576675348869241625089131927267773629264112554863560205741683008437015671577045348143356147368596086721006878157243247966695725069396786691105264470650407551586179024582131526208800833063775474840830644617972725190007938358148862068999884312511426305579307595236885625014057672886684815298848437514636781968698274934656250012724739427554787861210937590670660236629899207031250515806275286031052832031252252860979201801718750000070937666708078122558593751433522964707028808593750139628860198736572265625870783126313900412592193536
Technical details
Node id
n-f03e5c22b4dd0de2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8e69101448399e5f

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        33,          45;
  2:       36,       396,        1629,         2970,          2025;
  3:      216,      3564,       24462,        89397,        183465,         200475,          91125;
  4:     1296,     28512,      274104,      1504008,       5151681,       11280060,       15418350,        12028500,          4100625;
  5:     7776,    213840,     2643840,     19352520,      92877030,      305366193,      696577725,      1088579250,       1115370000,        676603125,         184528125;
  6:    46656,   1539648,    23269680,    212984640,    1314868140,     5768002548,    18436017069,     43260019110,      73961332875,      89852895000,       73626721875,       36536568750,        8303765625;
  7:   279936,  10777536,   192525984,   2115091440,   15964837560,    87582972708,   360129252798,   1127508792597,    2700969395985,    4926542214825,     6735165845625,     6692281509375,     4568731846875,     1918169859375,      373669453125;
  8:  1679616,  73903104,  1523411712,  19528895232,  174250712160,  1147516590528,  5769431306352,  22590551869776,   69619422907521,  169429139023320,   324530510982300,   484108561629000,   551340143943750,   463429838025000,   271134555187500,    98648735625000,    16815125390625;
  9: 10077696, 498845952, 11654855424, 170771597568, 1758511342656, 13514173329888, 80325917372736, 377572790902032, 1422829157933574, 4330590624228033, 10671218684501805, 21238469488239300, 33887496391623000, 42759689051598750, 41730298463418750, 30393675446062500, 15557354011406250, 4994092241015625, 756680642578125;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+45*x+117*x^2+123*x^3+45*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 45*x + 117*x^2 + 123*x^3 + 45*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 45*x + 117*x^2 + 123*x^3 + 45*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.