Real-rooted atlas

A038281 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(14 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 84\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0196849289
127441764378274913
238416329281058415128183521
35378245762402469605292056702431419857
475295369680832518616014817602381402041272924137569
51054135041581202561016487363630312077792401000188714422187410338673
614757890562529924096189744307281318988821781872037340352400075224494465616975757441
720661046784398463045123415397529617076987648548903174411762210881680315841543147282668619683118587876497
8289254654976619831403520597694567680341539752960128077407360329341904645881105440720135360578680202755620590492015993900449
9404956516966495454036142081022721815808065746402444802817702961920845310888576181138047552277252113602970558360212182740909354617714734271896307633
1056693912375296145784346107904171817265055744122726617896960591717622003202028746132582450718653314569315671016961247634511201188223344076385786429760696531441582622237229761
1179371477325414422110625826365442842794749104128223362444572467211965845244953604615397451624961318684986178562825753541811245413896207685406416215204634071041627082233369672226215943239904578032905937
12111120068255580163333602047667404846432314235367424397991264874577922345305668010905610051310005761024323077821613747279121099170713614835206094508821193151563584227069481038417693725795294787816763124873084782969168377826559400929
1315556809555781222450004030715011072075006046072516608069648471353051136044774017298390016021107751012098150475384825043207680207692885323123204450561828352640741760304725440953691820361289289206042480663514717320328111673401004423490143489072862423051509815793
1421779533378093711367467268586774986752120009673716026572801200096737160265728083578165623661363204298305660645441536168862008096785203251692451458199552012461573119387392023736329751214080356044946268211241615643070310437156824169920244990049472011249543088032141551684304672148661191875666868481
Technical details
Node id
n-656bbcfcd6756619 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f0770160d0c19011

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:           196,            84,              9;
  1:          2744,          1764,            378,            27;
  2:         38416,         32928,          10584,          1512,            81;
  3:        537824,        576240,         246960,         52920,          5670,          243;
  4:       7529536,       9680832,        5186160,       1481760,        238140,        20412,          729;
  5:     105413504,     158120256,      101648736,      36303120,       7779240,      1000188,        71442,        2187;
  6:    1475789056,    2529924096,     1897443072,     813189888,     217818720,     37340352,      4000752,      244944,       6561;
  7:   20661046784,   39846304512,    34153975296,   17076987648,    5489031744,   1176221088,    168031584,    15431472,     826686,     19683;
  8:  289254654976,  619831403520,   597694567680,  341539752960,  128077407360,  32934190464,   5881105440,   720135360,   57868020,   2755620,   59049;
  9: 4049565169664, 9545403614208, 10227218158080, 6574640244480, 2817702961920, 845310888576, 181138047552, 27725211360, 2970558360, 212182740, 9093546, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (14+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 84*x + 9*x^2.
%t Mathematica
P[0, x_] := 196 + 84*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 84*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 3*x)*v[i - 1]); 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.