Real-rooted atlas

A367270 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(4 + 2\,x - 11\,n - 5\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2} - 4\,n - 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2527
32518346
4129132424307
56819004208052086
63653591036301040135614374
71982537926288751106022052107100108

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 4\,x - 2\,x^{2} + 5\,n + 10\,n x + 5\,n x^{2} - 2\,n^{2} - 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(10 + 4\,x - 24\,n - 8\,n x + 12\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-c7b2a4255e084d17 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-8d798a959a00be52)
Fingerprint
f6-b98f707c8be2e8fb

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:      1;
  2:      5,       2;
  3:     25,      18,       3;
  4:    129,     132,      42,      4;
  5:    681,     900,     420,     80,      5;
  6:   3653,    5910,    3630,   1040,    135,     6;
  7:  19825,   37926,   28875,  11060,   2205,   210,    7;
  8: 108545,  239624,  217644, 104440,  28210,  4200,  308,   8;
  9: 598417, 1497096, 1579536, 911232, 311220, 63504, 7392, 432, 9;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (7*n^2-11*n+4+(3*n^2-5*n+2)*x)*P(n-1,x) - (4*n-2+(6*n-3)*x+(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 - 11*n + 4 + (3*n^2 - 5*n + 2)*x)*P[n - 1, x] - (4*n - 2 + (6*n - 3)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((7*n^2 - 11*n + 4 + (3*n^2 - 5*n + 2)*x)*v[i - 1] - (4*n - 2 + (6*n - 3)*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