Real-rooted atlas

A013610 → Veronese 2 (j=1) → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 2\,n + 9\,n x - 18\,n^{2} x + 9\,n^{3} x\right)\,P_{n}(x) = \left(72\,x - 2\,n - 36\,n x - 162\,n x^{2} - 9\,n^{2} x - 81\,n^{2} x^{2} + 9\,n^{3} x + 81\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 36\,x^{2} + 18\,n x^{2} - 162\,n x^{3} + 18\,n^{2} x^{2} - 162\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3Pn(1)
00
133
266
395463
412216228
5155407291284
618108043745472
721189015309874825968

All 15 rows

Technical details
Node id
n-dfae0097fdfbe5b8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-56d80b07a9b89537

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half affine: slope 1, intercept -1)
  • 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:  3;
  2:  6;
  3:  9,   54;
  4: 12,  216;
  5: 15,  540,   729;
  6: 18, 1080,  4374;
  7: 21, 1890, 15309,   8748;
  8: 24, 3024, 40824,  69984;
  9: 27, 4536, 91854, 314928, 98415;
%F Formula
The row polynomials satisfy -(2*n-2-(9*n^3-18*n^2+9*n)*x)*P(n,x) = -(2*n-(9*n^3-9*n^2-36*n+72)*x-(81*n^3-81*n^2-162*n)*x^2)*P(n-1,x) - (4*x-(18*n^2+18*n+36)*x^2+(162*n^2+162*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - (9*n^3 - 9*n^2 - 36*n + 72)*x - (81*n^3 - 81*n^2 - 162*n)*x^2)*P[n - 1, x] - (4*x - (18*n^2 + 18*n + 36)*x^2 + (162*n^2 + 162*n)*x^3)*D[P[n - 1, x], x])/(-2*n + 2 + (9*n^3 - 18*n^2 + 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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - (9*n^3 - 9*n^2 - 36*n + 72)*x - (81*n^3 - 81*n^2 - 162*n)*x^2)*v[i - 1] - (4*x - (18*n^2 + 18*n + 36)*x^2 + (162*n^2 + 162*n)*x^3)*deriv(v[i - 1], x))/(-2*n + 2 + (9*n^3 - 18*n^2 + 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.