Real-rooted atlas

A347823 → reverse → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(1 - 12\,x^{2} - 26\,x^{3} - 21\,x^{4} - 6\,x^{5} + 2\,n + 14\,n x + 36\,n x^{2} + 44\,n x^{3} + 26\,n x^{4} + 6\,n x^{5}\right)\,P_{n-1}(x) + \left(-2\,x - 13\,x^{2} - 33\,x^{3} - 41\,x^{4} - 25\,x^{5} - 6\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
25181639
37397244162
4968192240112621
5111054007607202722268
6131507201840264020166408019
71520311763780728084005376147227702

All 15 rows

Technical details
Node id
n-c1e56615ee6819d6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-67d96af909b5ec1e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  3,   5;
  2:  5,  18,   16;
  3:  7,  39,   72,    44;
  4:  9,  68,  192,   240,   112;
  5: 11, 105,  400,   760,   720,   272;
  6: 13, 150,  720,  1840,  2640,  2016,   640;
  7: 15, 203, 1176,  3780,  7280,  8400,  5376,  1472;
  8: 17, 264, 1792,  6944, 16800, 25984, 25088, 13824,  3328;
  9: 19, 333, 2592, 11760, 34272, 66528, 86016, 71424, 34560, 7424;
%F Formula
The row polynomials satisfy (2*n-1+(6*n-3)*x+(6*n-3)*x^2+(2*n-1)*x^3)*P(n,x) = (2*n+1+14*n*x+(36*n-12)*x^2+(44*n-26)*x^3+(26*n-21)*x^4+(6*n-6)*x^5)*P(n-1,x) - (2*x+13*x^2+33*x^3+41*x^4+25*x^5+6*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + 14*n*x + (36*n - 12)*x^2 + (44*n - 26)*x^3 + (26*n - 21)*x^4 + (6*n - 6)*x^5)*P[n - 1, x] - (2*x + 13*x^2 + 33*x^3 + 41*x^4 + 25*x^5 + 6*x^6)*D[P[n - 1, x], x])/(2*n - 1 + (6*n - 3)*x + (6*n - 3)*x^2 + (2*n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + 14*n*x + (36*n - 12)*x^2 + (44*n - 26)*x^3 + (26*n - 21)*x^4 + (6*n - 6)*x^5)*v[i - 1] - (2*x + 13*x^2 + 33*x^3 + 41*x^4 + 25*x^5 + 6*x^6)*deriv(v[i - 1], x))/(2*n - 1 + (6*n - 3)*x + (6*n - 3)*x^2 + (2*n - 1)*x^3)); 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.