Real-rooted atlas

A038335 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 3\,n x\right)\,P_{n}(x) = \left(48 + 72\,x + 27\,x^{2} + 36\,n x + 27\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121830
2144432243819
3172877768748291621168
42073612441620995213996832805527877
5248832186624041990404199040196830035429412835746
62985984268738567558272010077696070858800255091683720087306307575
7358318083762339841269789696211631616019840464001071385056312487308382637527204354164
8429981696515978035220316635136406332702724761711360034284321792149993907843673320192387420489167501234313
951597803526965703475231345665638473139886489610285296537609256766883845399780682241983592903684184141281238742048903857931654822
1061917364224928760463360470184984576012538266255360205705930752002221624052121616199342046720793437161472025104847687204649045868003835462841188165085170491
1174300837068812259638116352689604644044802068813932134403879026122752004887572914667524276626300334082618342632857601104613298236803068370272880050628109502523765727153082001585717384120
12891610044825616048980806860899303068742451233101022914150406982247020953600100544357101731841026390312080179275408267826298883976607873652480147281773098240036452238841814454226471004352367158397425345185797569946509
131069932053790722086367504891904140829806580203525163759574607462412102561502986240019606149634837708822873841240643993619606149634837708812407016565795737657439891508313600189551641977434884229664738339456572767099983468355861215966061015104196571356458
1412839184645488642696228775552614419716172921228492878864691684913971220332303325016883203659814598503038976480350666053523865647054759123610501123473964638422806528192998035467933696079611689630522649623686122534700953648112436398611312597846842822980834315188682441522707439527650560767
Technical details
Node id
n-bd4f560f4225207a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b0ee4e6a5f8b87bd

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,          18;
  2:        144,         432,          243;
  3:       1728,        7776,         8748,         2916;
  4:      20736,      124416,       209952,       139968,         32805;
  5:     248832,     1866240,      4199040,      4199040,       1968300,       354294;
  6:    2985984,    26873856,     75582720,    100776960,      70858800,     25509168,      3720087;
  7:   35831808,   376233984,   1269789696,   2116316160,    1984046400,   1071385056,    312487308,     38263752;
  8:  429981696,  5159780352,  20316635136,  40633270272,   47617113600,  34284321792,  14999390784,   3673320192,   387420489;
  9: 5159780352, 69657034752, 313456656384, 731398864896, 1028529653760, 925676688384, 539978068224, 198359290368, 41841412812, 3874204890;
%F Formula
The row polynomials satisfy (4+3*n*x)*P(n,x) = (48+(36*n+72)*x+(27*n+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[((48 + (36*n + 72)*x + (27*n + 27)*x^2)*P[n - 1, x])/(4 + 3*n*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] = ((48 + (36*n + 72)*x + (27*n + 27)*x^2)*v[i - 1])/(4 + 3*n*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.