Real-rooted atlas

A008459 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2} x\right)\,P_{n}(x) = \left(-4 - 8\,x - 3\,x^{2} + 8\,n + 10\,n x + 2\,n x^{2} + 4\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 7\,x^{2} - 3\,x^{3} - 2\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2612321
32060364120
470280270805705
52521260168084015064188
69245544945067202100252724997
734322402449896462002100045363928149488

All 15 rows

Technical details
Node id
n-7a0ff194f56f2d4f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-59bab5701f3316b8

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     2,      2;
  2:     6,     12,       3;
  3:    20,     60,      36,       4;
  4:    70,    280,     270,      80,       5;
  5:   252,   1260,    1680,     840,     150,      6;
  6:   924,   5544,    9450,    6720,    2100,    252,      7;
  7:  3432,  24024,   49896,   46200,   21000,   4536,    392,     8;
  8: 12870, 102960,  252252,  288288,  173250,  55440,   8820,   576,   9;
  9: 48620, 437580, 1235520, 1681680, 1261260, 540540, 129360, 15840, 810, 10;
%F Formula
The row polynomials satisfy (2*n+n^2*x)*P(n,x) = (8*n-4+(4*n^2+10*n-8)*x+(3*n^2+2*n-3)*x^2)*P(n-1,x) - (4*x+(2*n+7)*x^2+(2*n+3)*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[((8*n - 4 + (4*n^2 + 10*n - 8)*x + (3*n^2 + 2*n - 3)*x^2)*P[n - 1, x] - (4*x + (2*n + 7)*x^2 + (2*n + 3)*x^3)*D[P[n - 1, x], x])/(2*n + n^2*x)]];
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] = ((8*n - 4 + (4*n^2 + 10*n - 8)*x + (3*n^2 + 2*n - 3)*x^2)*v[i - 1] - (4*x + (2*n + 7)*x^2 + (2*n + 3)*x^3)*deriv(v[i - 1], x))/(2*n + n^2*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023