Real-rooted atlas

A038322 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(19 + 46\,x + 27\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1192746
236110267292116
3685929241415531968397336
4130321740772157901414959085314414477456
524760991759333550002110710556305048689514348907205962976
64704588140112803814250601352700113940287775301516357753983874204899474296896
7893871739889167150937906599591897787885051275803836651087790639675152692503710460353203435817657216
8169835630411930762956249603005229722729275170552484805457927055114725743283916046302812158997368685628242953648120047612231936
932268769777941270058189632345876991831677784342360732165803466611034235615452552522223214639260284135942750226188482954507382517625597484987922190162669056
106131066257801871256784003305571457855600052111289292648440525044310934941089533871969958361060269536486349086097075143252404588067820133845144886352214753020589113209464942420747482776576
11116490258898219182092667856689712938163242449005551574327704404951567632299791466703118763206953549544431926662512938824498572176234937903196353914693245351514062380644168854303124660778164155590605665555231951354384207722496
122213314919066161377428438830229562949901219278373141397321630184492540446775205440568009510158257302648704216168413213175491675162051348912363131682418219217313751498495115068740928956832604905562113287107074126746580917465924415009463529699912189762301673555234816
13420529834622570597768735365922225116623869101470528778345138442655569657381226149730486892203853136361942192787426695942580522049491966368444719689228225425889000293352993240256537105494752363084413054038912806606384824261565320274330704166343707337491835878288740525551530189762674129065876983540801536
14799006685782884121158960277487331683021468290984159300545798346075067852866260283261558283095133262241926969196248090506110619759080235814560788147320899348190672566183443990130053160336513728337801232082571609077426268587701639324590813291387917309906557694154849307588641417181239711077979670703047687022109418989131512359209189937030341242876870656
Technical details
Node id
n-e7cce546ab56eb06 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5254523500bc8159

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:           19,            27;
  2:          361,          1026,            729;
  3:         6859,         29241,          41553,          19683;
  4:       130321,        740772,        1579014,        1495908,          531441;
  5:      2476099,      17593335,       50002110,       71055630,        50486895,        14348907;
  6:     47045881,     401128038,     1425060135,     2700113940,      2877753015,      1635775398,       387420489;
  7:    893871739,    8891671509,    37906599591,    89778788505,    127580383665,    108779063967,     51526925037,     10460353203;
  8:  16983563041,  193076295624,   960300522972,  2729275170552,   4848054579270,   5511472574328,   3916046302812,   1589973686856,   282429536481;
  9: 322687697779, 4127005818963, 23458769918316, 77784342360732, 165803466611034, 235615452552522, 223214639260284, 135942750226188, 48295450738251, 7625597484987;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (19+46*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[((19 + 46*x + 27*x^2)*P[n - 1, x])/(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] = ((19 + 46*x + 27*x^2)*v[i - 1])/(1 + 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.