Real-rooted atlas

A038303 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114620
219616836400
32744352815122168000
4384166585642336120961296160000
553782411524809878404233609072077763200000
6752953619361664207446401185408038102406531844665664000000
71054135043162405124065949442904249601244678403200601645722882799361280000000
814757890565059848192758977228865055191043485099520119489126425604812831352832167961625600000000
92066104678479692609024136615901184136615901184878245079043763907481610754021376197522841621163161610077696512000000000
1028925465497612396628070402390778270720273231802368020492385177601053894094848376390748160921773260801481421312014108774406046617610240000000000
11404956516966419090807228416409088726323205259712195584045083247390720270499484344321159283504332835488270540807604629401601086375628809311791104362797056204800000000000
12566939123752962915686922158086872690602229769818129431756809467481952051206491987624263683245993812131841192405890170883193944348672060837035212807821904527366094990540821767823364096000000000000
137937147732541444422125165273088113711789964165121786899556579737619145352391925760147692718451998728439583911542784361696453351833611625957429166082768085102182404745288746598455464413921283961743851521306069401681920000000000000
14111120068255580166667204095334809618572925694146969631839301189966233637524890688174489632164192018435276820676980583279820810127500693851340837978127601942528108508936005550082325191485833216362367504285696388250897448962559896027136783641640961638400000000000000
Technical details
Node id
n-03dee8d451c5f864 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-5184fe74b2189306

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,           6;
  2:         196,         168,           36;
  3:        2744,        3528,         1512,          216;
  4:       38416,       65856,        42336,        12096,        1296;
  5:      537824,     1152480,       987840,       423360,       90720,        7776;
  6:     7529536,    19361664,     20744640,     11854080,     3810240,      653184,       46656;
  7:   105413504,   316240512,    406594944,    290424960,   124467840,    32006016,     4572288,     279936;
  8:  1475789056,  5059848192,   7589772288,   6505519104,  3485099520,  1194891264,   256048128,   31352832,   1679616;
  9: 20661046784, 79692609024, 136615901184, 136615901184, 87824507904, 37639074816, 10754021376, 1975228416, 211631616, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (14+6*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 + 6*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 + 6*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.