Real-rooted atlas

A038322 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
127835
2729432641225
31968317496518451242875
45314416298562799365529640961500625
514348907212576401259712037324805529603276852521875
63874204896887475365101833602015539204478976053084162621441838265625
71046035320321695547384192849310089523422720282175488050164531249545216209715264339296875
8282429536481669462604992694257516288411411861504152374763520361184624645350883328452984832167772162251875390625
976255974849872033492662663224100653779712166621803909127405413507072219419659468843342154956855037657088407686348813421772878815638671875
102058911320946496100477987989608133970650652806426841007923203332436078182401184866161131522925595459584049533891379205503765708803623878656010737418242758547353515625
1155590605665555231811841962432911226842103147154240238596472419148801413905021743104058650874976010241737803702992896367789143490560544872805171205381459804160318901321728858993459296549157373046875
1215009463529699912153366981438933020886968414196779737685894730070893568057263153380595712027146976417467596893841399961616384238327364981882884413469721886720581197658849280516620141199362783138807808687194767363379220508056640625
13405255515301897626715609842070887908584277508303482451708163014905025488364236822332629818432327680119107359031639080964705475912361050112139421508514401484830982557447644774451000094564024320604445565203251248844086077030424120536334336549755813888118272717781982421875
1410941898913151235920945388617713812534190487415115596972288070410360309996678197104648441734071367419863045002509079328841400322223337368590596177927528761459777680179219519011192016207872385560714904023859257120105911707238461543548457058304455878136718950420780769764966443980465111044139545122369384765625
Technical details
Node id
n-3c6b4dd8de6c9824 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-036b202913a03c7f

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:            27,              8;
  2:           729,            432,             64;
  3:         19683,          17496,           5184,            512;
  4:        531441,         629856,         279936,          55296,          4096;
  5:      14348907,       21257640,       12597120,        3732480,        552960,         32768;
  6:     387420489,      688747536,      510183360,      201553920,      44789760,       5308416,       262144;
  7:   10460353203,    21695547384,    19284931008,     9523422720,    2821754880,     501645312,     49545216,     2097152;
  8:  282429536481,   669462604992,   694257516288,   411411861504,  152374763520,   36118462464,   5350883328,   452984832,   16777216;
  9: 7625597484987, 20334926626632, 24100653779712, 16662180390912, 7405413507072, 2194196594688, 433421549568, 55037657088, 4076863488, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (27+8*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[(27 + 8*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] = (27 + 8*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

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.