Real-rooted atlas

A038303 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114317
2196849289
327441764378274913
438416329281058415128183521
55378245762402469605292056702431419857
675295369680832518616014817602381402041272924137569
71054135041581202561016487363630312077792401000188714422187410338673
814757890562529924096189744307281318988821781872037340352400075224494465616975757441
920661046784398463045123415397529617076987648548903174411762210881680315841543147282668619683118587876497
10289254654976619831403520597694567680341539752960128077407360329341904645881105440720135360578680202755620590492015993900449
11404956516966495454036142081022721815808065746402444802817702961920845310888576181138047552277252113602970558360212182740909354617714734271896307633
1256693912375296145784346107904171817265055744122726617896960591717622003202028746132582450718653314569315671016961247634511201188223344076385786429760696531441582622237229761
1379371477325414422110625826365442842794749104128223362444572467211965845244953604615397451624961318684986178562825753541811245413896207685406416215204634071041627082233369672226215943239904578032905937
14111120068255580163333602047667404846432314235367424397991264874577922345305668010905610051310005761024323077821613747279121099170713614835206094508821193151563584227069481038417693725795294787816763124873084782969168377826559400929
Technical details
Node id
n-087fd98ce32430ab · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f0770160d0c19011

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:          14,           3;
  2:         196,          84,           9;
  3:        2744,        1764,         378,          27;
  4:       38416,       32928,       10584,        1512,         81;
  5:      537824,      576240,      246960,       52920,       5670,        243;
  6:     7529536,     9680832,     5186160,     1481760,     238140,      20412,       729;
  7:   105413504,   158120256,   101648736,    36303120,    7779240,    1000188,     71442,     2187;
  8:  1475789056,  2529924096,  1897443072,   813189888,  217818720,   37340352,   4000752,   244944,   6561;
  9: 20661046784, 39846304512, 34153975296, 17076987648, 5489031744, 1176221088, 168031584, 15431472, 826686, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (14+3*x)*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[(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] = 1; 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.