Real-rooted atlas

A038320 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(17 + 12\,x + 6\,n x\right)\,P_{n}(x) = \left(289 + 408\,x + 108\,x^{2} + 102\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 408\,x + 108\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0289408108805
1491310404550886421689
283521235824187272587526480571849
314198575011260530604024969605508004665614831573
42413756910222970413530402084896640280908004758912326592379744237
541033867320275557963220235676252567504011142684002831552643886444822394889622332785
669757574413939251260872992008656686983610883788512560012836371968264278246430457036815116544241742606737
71185878764977533818036281595396760624175180820774411592848433604909912277761347819056642329963315223128312321007769606029945866637
8201599390044914230545179640339021811632604254391361664032846403895200166937017443845728230990720132031254528019659065472017132083200665127936149495670901429
9342718963076332661111948592687044119863921809944639807889608774682183432005202870376999682142358390529286172461149184012254150810880160184977920012437892403243535646723686959498014761
1058262223722976149351530682991521437000452240047222541183564549760223754395677516001516265081297049672840185277995522518364148867072624961691354880108925784985600126866502512648881271930882829817036890516927001823065
11990457803290593790889069007842716288706454495500392498160156776549696549441349385900400418868228708309952229966870663385856927597461499371522762330675788569660181496204544009345832351764489813805483622462538956513281828497162242213377067634052421
121683778265594009291663969109528197416572601134749409110810778374301165347968130767041153844295201107673760361975206468415144022357292163153831369097863168109572450139613260828646392193362944055607702492998656778561901700710474421358250803243518232461312117546246144053931890727202009981
132862423051509815793303080087806921672201123179148931533255802290404538997636443203031417772202754116002824568088923036776321938429080633456611201005283748899943884803991567826514483072012174716682179251200283599282714293145649633321233420288063258154513182720554857463881728029974292766720075229597532161309990504672878728417
144866119187566686848154958522588988463225621821766322098360398404792231035441208558080687121361699290932960069843865471551454832645272527099323001982464303819088556427485184013571330610149242444804730747053646794752001285650081638128926722700052675098063667243015545068964249605030707672527667204076503816273920020462450528747524795886842675231729593020262558227233
Technical details
Node id
n-a7921ed99a2ce929 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-dd409b60e0215f94

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,             408,             108;
  1:           4913,           10404,            5508,             864;
  2:          83521,          235824,          187272,           58752,            6480;
  3:        1419857,         5011260,         5306040,         2496960,          550800,           46656;
  4:       24137569,       102229704,       135304020,        84896640,        28090800,         4758912,          326592;
  5:      410338673,      2027555796,      3220235676,      2525675040,      1114268400,       283155264,        38864448,        2239488;
  6:     6975757441,     39392512608,     72992008656,     68698361088,     37885125600,     12836371968,      2642782464,      304570368,       15116544;
  7:   118587876497,    753381803628,   1595396760624,   1751808207744,   1159284843360,    490991227776,    134781905664,    23299633152,     2312831232,     100776960;
  8:  2015993900449,  14230545179640,  33902181163260,  42543913616640,  32846403895200,  16693701744384,   5728230990720,  1320312545280,   196590654720,   17132083200,    665127936;
  9: 34271896307633, 266111194859268, 704411986392180, 994463980788960, 877468218343200, 520287037699968, 214235839052928, 61724611491840, 12254150810880, 1601849779200, 124378924032, 4353564672;
%F Formula
The row polynomials satisfy (17+(6*n+12)*x)*P(n,x) = (289+(102*n+408)*x+(36*n+108)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 408*x + 108*x^2.
%t Mathematica
P[0, x_] := 289 + 408*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((289 + (102*n + 408)*x + (36*n + 108)*x^2)*P[n - 1, x])/(17 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 408*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (102*n + 408)*x + (36*n + 108)*x^2)*v[i - 1])/(17 + (6*n + 12)*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.