Real-rooted atlas

A027465 → Veronese 2 (j=1) → D

nonnegative coefficients

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

Recurrence

\(\left(-9 + 3\,n\right)\,P_{n}(x) = \left(-4\,x + 9\,n + n x\right)\,P_{n-1}(x) + \left(18\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
20
311
41212
590292
654036576
7283537833216
81360830247216704
96123620412972482624
102624401224729720120394752
11108256567359680190198051838336
1243302603464208577368237601808395776
13168880141688801437528922316603510637764096
14644815087881073222517352194594449140252167804928
Technical details
Node id
n-76fec8abefee5dcd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-bac3b0b37dbaf373

Observed on the stored rows ?

  • Degrees: -1, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 (half affine: slope 1, intercept -3)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     0;
  2:     0;
  3:     1;
  4:    12;
  5:    90,     2;
  6:   540,    36;
  7:  2835,   378,   3;
  8: 13608,  3024,  72;
  9: 61236, 20412, 972, 4;
%F Formula
The row polynomials satisfy (3*n-9)*P(n,x) = (9*n+(n-4)*x)*P(n-1,x) + (18*x-2*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + (n - 4)*x)*P[n - 1, x] + (18*x - 2*x^2)*D[P[n - 1, x], x])/(3*n - 9)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = ((9*n + (n - 4)*x)*v[i - 1] + (18*x - 2*x^2)*deriv(v[i - 1], x))/(3*n - 9)); 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.