Real-rooted atlas

A038335 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(48 + 72\,x + 27\,x^{2} + 36\,n + 27\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121930
236037881819
356169720510372921168
482944202176174960612366561527877
511819523732480454896026244006889055904912835746
6164229126382540810077696081881280354294007440174531441306307575
722394880010346434562010500352211631616012896301604464104407812182747829697204354164
830098718721612431360037247164416482520084483809369088018570674304535692528080353879243046721167501234313
93998829772824379962163265303470080010056734392329771031710726171177922562507041031046198727824081358302693874204893857931654822
105262975959043598946795520109709829734401959104102400022627652382720175878570792969256766883840322333846848069735688020081358302690348678440188165085170491
11687282742886452103461994496178147866378240362042438123520484878265344000448027517177856290199641808384130917131642880398888135474407670925682200805447196631313810596092001585717384120
12891610044825607422653623173122813586947702784641332318961664097751458293350401047337053143040080644953092014084477365902186496176738127717888047866576256928082845997367760790802702146828242953648145185797569946509
1311501769578250241043183752445952043422523695562752109729890960408576187589703296286720228738412406439936204230725362892800134792278739509248654814763194775042297595660332544056003894220605768811801538207207710326345931325418658283291015104196571356458
14147650623423119361449222966859530246572057640409497601823745995213635584345649156525287014447272605230664253444803506660535238656367615305653207040021229783901472706569167406684726850562894970532019005446414991519814841692523916151175607473085535287262287679245496122707439527650560767
Technical details
Node id
n-e63fbd290efa0220 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-87f28ed98a2aba0a

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:          21,            9;
  2:         360,          378,           81;
  3:        5616,         9720,         5103,           729;
  4:       82944,       202176,       174960,         61236,         6561;
  5:     1181952,      3732480,      4548960,       2624400,       688905,        59049;
  6:    16422912,     63825408,    100776960,      81881280,     35429400,      7440174,       531441;
  7:   223948800,   1034643456,   2010500352,    2116316160,   1289630160,    446410440,     78121827,     4782969;
  8:  3009871872,  16124313600,  37247164416,   48252008448,  38093690880,  18570674304,   5356925280,   803538792,   43046721;
  9: 39988297728, 243799621632, 653034700800, 1005673439232, 977103171072, 617117792256, 250704103104, 61987278240, 8135830269, 387420489;
%F Formula
The row polynomials satisfy (3*n+1+3*x)*P(n,x) = (36*n+48+(27*n+72)*x+27*x^2)*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[Together[((36*n + 48 + (27*n + 72)*x + 27*x^2)*P[n - 1, x])/(3*n + 1 + 3*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] = ((36*n + 48 + (27*n + 72)*x + 27*x^2)*v[i - 1])/(3*n + 1 + 3*x)); 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.