Real-rooted atlas

A133087 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 3\,x - 7\,n - 2\,n x\right)\,P_{n-1}(x) + \left(21 + 13\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
296116
3363310180
48110854121256
53245133241021611280
6729145812155401351814096
72916656163183375108020722120480
865611749620412136085670151225224165536
92624476545991447484436288117182520348281327680
1059049196830295245262440153090612361701032404053011048576
11236196846369137781013450058748003980341292762997048605253415242880
125314412125764389723443302603247695173210467359619245640095594059436116777216
1321257649034497177147002121827417321040101761114426488144342035283663855831673840183886080
14478296922320522483611316448150859108049394053661970268375057842189187486486810819828819421268435456
Technical details
Node id
n-d8b2fe687f6a4e6e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-69bbf457349d7fd6

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:     4,     1;
  2:     9,     6,     1;
  3:    36,    33,    10,     1;
  4:    81,   108,    54,    12,     1;
  5:   324,   513,   324,   102,    16,     1;
  6:   729,  1458,  1215,   540,   135,    18,    1;
  7:  2916,  6561,  6318,  3375,  1080,   207,   22,   1;
  8:  6561, 17496, 20412, 13608,  5670,  1512,  252,  24,  1;
  9: 26244, 76545, 99144, 74844, 36288, 11718, 2520, 348, 28, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(7*n-11+(2*n-3)*x)*P(n-1,x) + (21+13*x+2*x^2)*d/dx 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[-(7*n - 11 + (2*n - 3)*x)*P[n - 1, x] + (21 + 13*x + 2*x^2)*D[P[n - 1, x], 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] = -(7*n - 11 + (2*n - 3)*x)*v[i - 1] + (21 + 13*x + 2*x^2)*deriv(v[i - 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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.