Real-rooted atlas

A038335 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(30 + 27\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1302757
290016207293249
327000729006561019683185193
481000029160003936600236196053144110556001
5243000001093500001968300001771470007971615014348907601692057
672900000039366000008857350000106288200007174453500258280326038742048934296447249
72187000000013778100000037200870000055801305000050221174500027119434230081358302690104603532031954897493193
865610000000047239200000001488034800000026784626400000301327047000002169554738400097629963228002510484768720282429536481111429157112001
9196830000000001594323000000005739562800000001205308188000000162716605380000014644494484200008786696690520003389154437772007625597484987076255974849876351461955384057
105904900000000005314410000000000215233605000000005165606520000000081358302690000000878669669052000006590022517890000033891544377720000114383962274805002287679245496100205891132094649362033331456891249
111771470000000000017537553000000000078918988500000000021308126895000000003835462841100000000483268317978600000043494148618074000002796052411161900000125822358502285500037746707550685650067944073591234170555906056655552320635899893042801193
1253144100000000000057395628000000000002841083586000000000085232507580000000000172595827849500000000248537992103280000000260964891708444000000201315773603656800000113240122652056950000452960490608227800001222993324642215060020012618039599882801500946352969991211176246293903439668001
131594323000000000000018653579100000000000010072932714000000000003324067795620000000000747915254014500000000012116227115034900000000145394725380418800000001308552528423769200000088327295668604421000004416364783430221050000158989132203487957800039024605177219771460058536907765829657190405255515301897626767046038752496061076057
14478296900000000000000602654094000000000000035255264499000000000000126918952196400000000000314124406686090000000000565423932034962000000000763322308247198700000000785131517054261520000000618291069680230947000000370974641808138568200000166938588813662355690000546344472481076800440001229275063082422800990017020731642679700321401094189891315123592093821624208892275481335249
Technical details
Node id
n-fd7d4197f6940d35 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-249d425e0def4c4f

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:             30,              27;
  2:            900,            1620,             729;
  3:          27000,           72900,           65610,            19683;
  4:         810000,         2916000,         3936600,          2361960,           531441;
  5:       24300000,       109350000,       196830000,        177147000,         79716150,         14348907;
  6:      729000000,      3936600000,      8857350000,      10628820000,       7174453500,       2582803260,       387420489;
  7:    21870000000,    137781000000,    372008700000,     558013050000,     502211745000,     271194342300,     81358302690,     10460353203;
  8:   656100000000,   4723920000000,  14880348000000,   26784626400000,   30132704700000,   21695547384000,   9762996322800,   2510484768720,   282429536481;
  9: 19683000000000, 159432300000000, 573956280000000, 1205308188000000, 1627166053800000, 1464449448420000, 878669669052000, 338915443777200, 76255974849870, 7625597484987;
%F Formula
The row polynomials satisfy P(n,x) = (30+27*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[(30 + 27*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] = (30 + 27*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.