Real-rooted atlas

A001263 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21012325
33560304129
4126280210605681
54621260126056010563653
617165544693042001260168719825
764352402436036277201155025202528108545

All 15 rows

Technical details
Node id
n-a60016fa379da581 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2c1fea4a558d4e6c

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:     3,      2;
  2:    10,     12,      3;
  3:    35,     60,     30,      4;
  4:   126,    280,    210,     60,      5;
  5:   462,   1260,   1260,    560,    105,      6;
  6:  1716,   5544,   6930,   4200,   1260,    168,     7;
  7:  6435,  24024,  36036,  27720,  11550,   2520,   252,    8;
  8: 24310, 102960, 180180, 168168,  90090,  27720,  4620,  360,   9;
  9: 92378, 437580, 875160, 960960, 630630, 252252, 60060, 7920, 495, 10;
%F Formula
The row polynomials satisfy (n^2+n)*P(n,x) = (4*n^2+2*n+(3*n^2+n)*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*x^2)*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[((4*n^2 + 2*n + (3*n^2 + n)*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n)]];
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] = ((4*n^2 + 2*n + (3*n^2 + n)*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n)); 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