Real-rooted atlas

A013610 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 - 3\,x - 8\,n + 3\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(-2\,n - 3\,n x + 9\,n x^{2} + 3\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2241842
36321681360
412075612963242496
5195180056706480121515360
62883510162003402029160437487552
739960483685511340017860512247215309473088

All 15 rows

Technical details
Node id
n-a698946f19b5cd71 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f45c55ca90e4ac8

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   3;
  2:  24,    18;
  3:  63,   216,     81;
  4: 120,   756,   1296,    324;
  5: 195,  1800,   5670,   6480,    1215;
  6: 288,  3510,  16200,  34020,   29160,    4374;
  7: 399,  6048,  36855, 113400,  178605,  122472,   15309;
  8: 528,  9576,  72576, 294840,  680400,  857304,  489888,   52488;
  9: 675, 14256, 129276, 653184, 1990170, 3674160, 3857868, 1889568, 177147;
%F Formula
The row polynomials satisfy (3*n^2-8*n+5+(3*n-3)*x)*P(n,x) = (3*n^2-2*n+(9*n^2-3*n)*x+9*n*x^2)*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[((3*n^2 - 2*n + (9*n^2 - 3*n)*x + 9*n*x^2)*P[n - 1, x])/(3*n^2 - 8*n + 5 + (3*n - 3)*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] = ((3*n^2 - 2*n + (9*n^2 - 3*n)*x + 9*n*x^2)*v[i - 1])/(3*n^2 - 8*n + 5 + (3*n - 3)*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

Same fingerprint ?

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.