Real-rooted atlas

A027465 → Veronese 2 (j=1) → θ+1

nonnegative coefficients

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

Recurrence

\(\left(54 - 54\,n + 3\,n x - 6\,n^{2} x + 3\,n^{3} x\right)\,P_{n}(x) = \left(72\,x - 162\,n - 36\,n x - 2\,n x^{2} - 9\,n^{2} x - n^{2} x^{2} + 9\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-324\,x + 36\,x^{2} + 18\,n x^{2} - 2\,n x^{3} + 18\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
266
327229
410824132
54051803588
614581080542592
751035670567411344
8174962721645369649344
9590491224723061812965213440
1019683052488018370812960150918528
1164953921651301010394106920247563934464
122125764866052051963127698242970021616782336
136908733337760282533202150038562895754212771314432
142232052212896301611821609830023136243243058968294302014464
Technical details
Node id
n-23c9241cd93998c4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f965cd08d5c2494d

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:     1;
  2:     6;
  3:    27,      2;
  4:   108,     24;
  5:   405,    180,     3;
  6:  1458,   1080,    54;
  7:  5103,   5670,   567,    4;
  8: 17496,  27216,  4536,   96;
  9: 59049, 122472, 30618, 1296, 5;
%F Formula
The row polynomials satisfy -(54*n-54-(3*n^3-6*n^2+3*n)*x)*P(n,x) = -(162*n-(9*n^3-9*n^2-36*n+72)*x-(n^3-n^2-2*n)*x^2)*P(n-1,x) - (324*x-(18*n^2+18*n+36)*x^2+(2*n^2+2*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(162*n - (9*n^3 - 9*n^2 - 36*n + 72)*x - (n^3 - n^2 - 2*n)*x^2)*P[n - 1, x] - (324*x - (18*n^2 + 18*n + 36)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(-54*n + 54 + (3*n^3 - 6*n^2 + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(162*n - (9*n^3 - 9*n^2 - 36*n + 72)*x - (n^3 - n^2 - 2*n)*x^2)*v[i - 1] - (324*x - (18*n^2 + 18*n + 36)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(-54*n + 54 + (3*n^3 - 6*n^2 + 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

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.