Real-rooted atlas

A038291 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + n x\right)\,P_{n}(x) = \left(81 + 18\,x + x^{2} + 9\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19211
281363120
37294868141300
4656158321458144514000
559049656102187032402256150000
653144170858829524558320607532471600000
747829697440174372008791854012757510206441817000000
8430467217652750444641044132269762296350244944158765769180000000
938742048977484097851656065217856417637200870496011642865223328729101900000000
103486784401774840978058113073352295825120558013050892820889644670699840328059001120000000000
11313810596097670925682263924380685284108358607891898850147315445219096446617321040108256544550108912210000000000
12282429536481753145430616690383311398340930030320106540634475227286686883437360388374134464292292551603800588061296132200000000000
13254186582832973431679485067343167948506398888135474413850282481753324067795625745302362872956220486839645674691115022934347581615211423000000000000
142287679245496171172243193212771032634593134569082279070417451355927005465369491386890488512214113132119686414363255907118216098072243171318427295823176415240000000000000
Technical details
Node id
n-2e0b1f4c9fd7aabe · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4905f7f4d43a1fb8

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:         9,         2;
  2:        81,        36,         3;
  3:       729,       486,        81,         4;
  4:      6561,      5832,      1458,       144,        5;
  5:     59049,     65610,     21870,      3240,      225,       6;
  6:    531441,    708588,    295245,     58320,     6075,     324,      7;
  7:   4782969,   7440174,   3720087,    918540,   127575,   10206,    441,     8;
  8:  43046721,  76527504,  44641044,  13226976,  2296350,  244944,  15876,   576,   9;
  9: 387420489, 774840978, 516560652, 178564176, 37200870, 4960116, 428652, 23328, 729, 10;
%F Formula
The row polynomials satisfy (9+n*x)*P(n,x) = (81+(9*n+18)*x+(n+1)*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[((81 + (9*n + 18)*x + (n + 1)*x^2)*P[n - 1, x])/(9 + 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] = ((81 + (9*n + 18)*x + (n + 1)*x^2)*v[i - 1])/(9 + 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

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.