Real-rooted atlas

A347823 → reverse → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-2 - 4\,x - x^{2} + 7\,n + 6\,n x + n x^{2}\right)\,P_{n-1}(x) + \left(-6 - 11\,x - 6\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
21614333
34460274135
4112208144445513
52726406002806561863
66401824216013604809076561
714724928705656002660756119822599

All 15 rows

Technical details
Node id
n-71321756900afd5a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f3e4733f729912cf

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:    5,     2;
  2:   16,    14,     3;
  3:   44,    60,    27,     4;
  4:  112,   208,   144,    44,     5;
  5:  272,   640,   600,   280,    65,     6;
  6:  640,  1824,  2160,  1360,   480,    90,    7;
  7: 1472,  4928,  7056,  5600,  2660,   756,  119,    8;
  8: 3328, 12800, 21504, 20608, 12320,  4704, 1120,  152,   9;
  9: 7424, 32256, 62208, 69888, 50400, 24192, 7728, 1584, 189, 10;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (7*n-2+(6*n-4)*x+(n-1)*x^2)*P(n-1,x) - (6+11*x+6*x^2+x^3)*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[((7*n - 2 + (6*n - 4)*x + (n - 1)*x^2)*P[n - 1, x] - (6 + 11*x + 6*x^2 + x^3)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((7*n - 2 + (6*n - 4)*x + (n - 1)*x^2)*v[i - 1] - (6 + 11*x + 6*x^2 + x^3)*deriv(v[i - 1], x))/(2*n - 1)); 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.