Real-rooted atlas

A038291 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114418
219611216324
327442352672645832
43841643904188163584256104976
55378247683204390401254401792010241889568
67529536129077769219840351232075264086016409634012224
71054135042108270081807088648605184024586240421478440140816384612220032
814757890563373232128337323212819275612166884147201573519362247884818350086553611019960576
9206610467845312840601660718178304404787855361734805094449565859849441116161156055048257536262144198359290368
1028925465497682644187136010625681203208095757107204047878553601387844075523304390656053949235205780275203670016010485763570467226624
114049565169664127272048189441818172116992015584332431360890533281792035621331271681017752322048207704555520296720793602825912320161480704419430464268410079232
125669391237529619437912814387230545291565465629090753871872018701198917632085491195052032284970650173446978873065472124622733312015825108992013564379136704643072167772161156831381426176
13793714773254144294808344351539250538573317406725294517204680704378179800334336019449246874337287409236904509442116924829859844536267492556872004245913608229056675846412251955230534533126710886420822964865671168
14111120068255580164444802730223206482546336418430976943386701924925447412324086552985642356137637445632181526304160481285927389523607552148184738090188828225664398131240322377711616418933794406429923842457613153337344268435456374813367582081024
Technical details
Node id
n-f038b923445dac3e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8d10fbbacee008e8

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,           4;
  2:         196,         112,          16;
  3:        2744,        2352,         672,          64;
  4:       38416,       43904,       18816,        3584,         256;
  5:      537824,      768320,      439040,      125440,       17920,       1024;
  6:     7529536,    12907776,     9219840,     3512320,      752640,      86016,      4096;
  7:   105413504,   210827008,   180708864,    86051840,    24586240,    4214784,    401408,     16384;
  8:  1475789056,  3373232128,  3373232128,  1927561216,   688414720,  157351936,  22478848,   1835008,   65536;
  9: 20661046784, 53128406016, 60718178304, 40478785536, 17348050944, 4956585984, 944111616, 115605504, 8257536, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (14+4*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 + 4*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 + 4*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.