Real-rooted atlas

A391533 → Veronese 2 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 2\,n\right)\,P_{n}(x) = \left(10 + 2\,x - n + n x + 3\,n^{2} - n^{2} x\right)\,P_{n-1}(x) + \left(24 + 26\,x - 6\,x^{2} - 12\,n - 8\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(12\,x + 8\,x^{2} - 4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 5 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0516
19514
21815134
33639782
47296291198
5144228979478
62885282904711154
75761200808191112786
81152268821446726916726
923045952548821523291316238
10460813056136646448133095139202
11921628416332801838448125191594642
12184326144079616504321607223681251228486
133686413209618764813414450528954876917551614
147372828262443673634790415148835168390615911331714
Technical details
Node id
n-f673dd234e42ccf6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b09a61eebcda220e

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
  • 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:    5,    1;
  1:    9,    5;
  2:   18,   15,    1;
  3:   36,   39,    7;
  4:   72,   96,   29,    1;
  5:  144,  228,   97,    9;
  6:  288,  528,  290,   47,   1;
  7:  576, 1200,  808,  191,  11;
  8: 1152, 2688, 2144,  672,  69,  1;
  9: 2304, 5952, 5488, 2152, 329, 13;
%F Formula
The row polynomials satisfy (2*n+6)*P(n,x) = (3*n^2-n+10-(n^2-n-2)*x)*P(n-1,x) - (12*n-24+(8*n-26)*x-(4*n-6)*x^2)*d/dx P(n-1,x) + (12*x+8*x^2-4*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 5 + x.
%t Mathematica
P[0, x_] := 5 + x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - n + 10 - (n^2 - n - 2)*x)*P[n - 1, x] - (12*n - 24 + (8*n - 26)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x] + (12*x + 8*x^2 - 4*x^3)*D[P[n - 1, x], {x, 2}])/(2*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 5 + x; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - n + 10 - (n^2 - n - 2)*x)*v[i - 1] - (12*n - 24 + (8*n - 26)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x) + (12*x + 8*x^2 - 4*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n + 6)); 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

Candidate equalities ?

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.