Real-rooted atlas

A038226 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 30\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 418\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121418361900
11331689711913685927000
214641101156262086301796130321810000
31610511390895480491082993907167655247609924300000
41771561183598147928101518258658023653261516342253447045881729000000
5194871712356176131220927631351479166560710037856291767559362253283789387173921870000000
6214358881296204999217906938588618603333041335620832701845585150641593914448287866071303216983563041656100000000
723579476913665536865125325527431610206954995162644529248746456782324783452599176793243893705295084168137274105932268769777919683000000000
8259374246014480100612903482260021845160395007066804848303622701010049211145234814464773618141014276919415308092475500758245354956467556906131066257801590490000000000
92853116706115420921741609468170514048052425974481885358380639119240302026590914289018350047521559012243187681231306703729845197249215214748662871924574185901719392111649025889821917714700000000000
1031384283767216505106089930861798507854342635580959067651801382805454674649538215714383735768770104547429826841140154784506496881230848915092240959448941166364678048962695134798786153767141745649082213314919066161531441000000000000
11345227121439317751918090500878033806021064538508807714667420742197124222427498656831058946092768531573213575463789116271736890307381756435202278971638091173377996466975372385233388846809207546610994350634813909223165040334264610234205298346225705915943230000000000000
1237974983358324191830414302856461031005106036615717123308005343890363383571302538347921116887008633142928183028436132767794048359782115867623986408903525160272044340031040422911828414694588985470602154490544156437666592044733996243708105738374987716476159453187587086799006685782884121478296900000000000000
13417724816941565110822870257122368513085834038157045559794548507347849245507535695380752188151928635642446858314975552132910074289088551233006139769744719665212973787778410451578528611630075281404100952965205298710981879553232805374692184527733513403945815610624324055342831548879759345951318361031541758799651518112702987479829914348907000000000000000
14459497298635721611269883443502357904164507627908260001201326031182533247282407443856865584365426203085817027914973303952977175392173074879584824112120066608341184560468542333112502993472707193781276070753031096086979355428855468466888819474918916489537621925897736158868674702582031344611753427921496240116015770775674774369202671878357257964500624288441413567621167681430467210000000000000000
Technical details
Node id
n-b92e5d1cc734bd04 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b53510a6d5d438b6

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:          121,           418,            361;
  1:         1331,          6897,          11913,            6859;
  2:        14641,        101156,         262086,          301796,          130321;
  3:       161051,       1390895,        4804910,         8299390,         7167655,          2476099;
  4:      1771561,      18359814,       79281015,       182586580,       236532615,        163422534,         47045881;
  5:     19487171,     235617613,     1220927631,      3514791665,      6071003785,       6291767559,       3622532837,        893871739;
  6:    214358881,    2962049992,    17906938588,     61860333304,    133562083270,     184558515064,     159391444828,      78660713032,      16983563041;
  7:   2357947691,   36655368651,   253255274316,   1020695499516,   2644529248746,    4567823247834,    5259917679324,    3893705295084,    1681372741059,     322687697779;
  8:  25937424601,  448010061290,  3482260021845,  16039500706680,  48483036227010,  100492111452348,  144647736181410,  142769194153080,   92475500758245,   35495646755690,   6131066257801;
  9: 285311670611, 5420921741609, 46817051404805, 242597448188535, 838063911924030, 2026590914289018, 3500475215590122, 4318768123130670, 3729845197249215, 2147486628719245, 741859017193921, 116490258898219;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+30*x+19*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 418*x + 361*x^2.
%t Mathematica
P[0, x_] := 121 + 418*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 30*x + 19*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] = 121 + 418*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 30*x + 19*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

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.