Real-rooted atlas

A038220 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-39 + 114\,x - 24\,x^{2} + 57\,n - 138\,n x + 24\,n x^{2} - 24\,n^{2} + 24\,n^{2} x + 6\,n^{3}\right)\,P_{n}(x) = \left(-81 + 378\,x + 48\,x^{2} - 32\,x^{3} + 99\,n - 426\,n x - 80\,n x^{2} + 32\,n x^{3} - 36\,n^{2} + 40\,n^{2} x + 32\,n^{2} x^{2} + 18\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(-162\,x - 324\,x^{2} + 320\,x^{3} - 64\,x^{4} - 36\,n x + 160\,n x^{2} - 64\,n x^{3} + 36\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
213417
3633699
429724816561
5132315602403123
65589918023526417185
7225995065219152134493747
88820926308813910417152256507809
9334611129470492534417280069122734371
101240029607986057153601503360108800102414648433
11450740727424980331024321174694413094403379278124995
121612037711951517618129355284167424133056006328324096415039057
135686418750561690494572878456043187211959833688657921597442197265619
1419822749320849020924728643920350269920097556659210315468834693121638411596679681
Technical details
Node id
n-956c16c9f5b0d912 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-81c406eb23fb5e9f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      3;
  2:     13,       4;
  3:     63,      36;
  4:    297,     248,     16;
  5:   1323,    1560,    240;
  6:   5589,    9180,   2352,     64;
  7:  22599,   50652,  19152,   1344;
  8:  88209,  263088, 139104,  17152,  256;
  9: 334611, 1294704, 925344, 172800, 6912;
%F Formula
The row polynomials satisfy (6*n^3-24*n^2+57*n-39+(24*n^2-138*n+114)*x+(24*n-24)*x^2)*P(n,x) = (18*n^3-36*n^2+99*n-81+(8*n^3+40*n^2-426*n+378)*x+(32*n^2-80*n+48)*x^2+(32*n-32)*x^3)*P(n-1,x) + ((36*n^2-36*n-162)*x-(16*n^2-160*n+324)*x^2-(64*n-320)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^3 - 36*n^2 + 99*n - 81 + (8*n^3 + 40*n^2 - 426*n + 378)*x + (32*n^2 - 80*n + 48)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] + ((36*n^2 - 36*n - 162)*x - (16*n^2 - 160*n + 324)*x^2 - (64*n - 320)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(6*n^3 - 24*n^2 + 57*n - 39 + (24*n^2 - 138*n + 114)*x + (24*n - 24)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^3 - 36*n^2 + 99*n - 81 + (8*n^3 + 40*n^2 - 426*n + 378)*x + (32*n^2 - 80*n + 48)*x^2 + (32*n - 32)*x^3)*v[i - 1] + ((36*n^2 - 36*n - 162)*x - (16*n^2 - 160*n + 324)*x^2 - (64*n - 320)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(6*n^3 - 24*n^2 + 57*n - 39 + (24*n^2 - 138*n + 114)*x + (24*n - 24)*x^2)); 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.