Real-rooted atlas

A027467 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118422
232414416484
3583238888646410648
410497693312311044608256234256
5188956820995209331202073602304010245153632
6340122244534963225194240746496012441601105924096113379904
7612220032952342272634894848235146240522547206967296516096163842494357888
8110199605761959104102415237476352677221171218811699203344302083715891223592966553654875873536
919835929036839671858073635263873843218284971622460949905408135444234242006581248191102976106168322621441207269217792
1035704672266247934371614720793437161472047018498457601828497162240487599243264902961561601146617856095551488047185920104857626559922791424
1164268410079232157100557971456174556175523840116370783682560517203483033601609077502771235757277839365675758387206306398208046714060802076180484194304584318301411328
1211568313814261763084883683803136377041339131494427928988083814401396449404190720496515343712256128726200221696245192762327043405455032320336341237760224227491849059696641677721612855002631049216
132082296486567116860155231834161152802069757788815366535383211612569636307684508958720145230738035834884303132978839552956251773075456159375295512576196759624089601748974436352105998450688392586854467108864282810057883082752
143748133675820810241166086032477585408168434649135651225614971968812057886729149536496257597444066460665003376641355486888334458883442506383071641666937624115281929916685054115841101853894901768903869857792494659436544169114337282684354566221821273427820544
Technical details
Node id
n-4675eb44920357da · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-0205823d5b324a40

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:           18,            4;
  2:          324,          144,           16;
  3:         5832,         3888,          864,           64;
  4:       104976,        93312,        31104,         4608,         256;
  5:      1889568,      2099520,       933120,       207360,       23040,        1024;
  6:     34012224,     45349632,     25194240,      7464960,     1244160,      110592,       4096;
  7:    612220032,    952342272,    634894848,    235146240,    52254720,     6967296,     516096,     16384;
  8:  11019960576,  19591041024,  15237476352,   6772211712,  1881169920,   334430208,   37158912,   2359296,    65536;
  9: 198359290368, 396718580736, 352638738432, 182849716224, 60949905408, 13544423424, 2006581248, 191102976, 10616832, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (18+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[(18 + 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] = (18 + 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.