Real-rooted atlas

A038271 → x ↦ x/(x+1) → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112719
214416849361
31728302417643436859
4207364838442336164642401130321
5248832725760846720493920144060168072476099
629859841045094415240960118540805186160121010411764947045881
735831808146313216256048128248935680145212480508243689882516823543893871739
84299816962006581248409677004847795650563485099520162637977647436076879060128576480116983563041
95159780352270888468486320730931286032171008752781496324391225395217076987648426924691262259850840353607322687697779
10619173642243611846246409481096396801474837217280150556299264010538940948485123096294401707698764803735591048048424328402824752496131066257801
117430083706884767637045248139056080486402433481408512028390616432640231856700866561352497421721656354059238401643660061120319600567440372867328681977326743116490258898219
1289161004482566241270313779220024075590041638935702536192051103109578752047696235606835232459938121318416229969060659259171762200320153408272371202684644766496284735050992138412872012213314919066161
1310699320537907281136514079129628397779927695366073969595645952885787232698368093007659433328647233929067036672421979195577139218461589806499845982922622476801396015278577922220933397737621592408033569688901040742052983462257059
14128391846454886410485334127149056397568918987735049276608109713817614881225509332582417361429760888012815191251040777011210127500693851340851692451458199552201026200115220485863264170027264124372270273305618137622748190416277353748376678223072849799006685782884121
Technical details
Node id
n-1afcf9ef6685c385 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-48a0cbb267bcb941

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:         12,           7;
  2:        144,         168,          49;
  3:       1728,        3024,        1764,         343;
  4:      20736,       48384,       42336,       16464,        2401;
  5:     248832,      725760,      846720,      493920,      144060,       16807;
  6:    2985984,    10450944,    15240960,    11854080,     5186160,     1210104,      117649;
  7:   35831808,   146313216,   256048128,   248935680,   145212480,    50824368,     9882516,     823543;
  8:  429981696,  2006581248,  4096770048,  4779565056,  3485099520,  1626379776,   474360768,   79060128,   5764801;
  9: 5159780352, 27088846848, 63207309312, 86032171008, 75278149632, 43912253952, 17076987648, 4269246912, 622598508, 40353607;
%F Formula
The row polynomials satisfy P(n,x) = (12+7*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[(12 + 7*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] = (12 + 7*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.