Real-rooted atlas

A132812 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 10\,n^{2} + 2\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(-12\,n x^{2} + 10\,n x^{3} + 12\,n^{2} x^{2} - 10\,n^{2} x^{3} + 48\,n^{3} x^{2} - 4\,n^{3} x^{3} - 48\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-3}(x) + \left(-4\,n x^{3} + 3\,n x^{4} + 4\,n^{2} x^{3} - 3\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{4} x^{3}\right)\,P_{n-3}'(x) + \left(-8\,n + 8\,n x - 6\,n x^{2} + 16\,n^{2} - 16\,n^{2} x + 12\,n^{2} x^{2} + 32\,n^{3} - 8\,n^{3} x + 6\,n^{3} x^{2} - 64\,n^{4} + 16\,n^{4} x - 12\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(8\,n x - 2\,n x^{2} - 3\,n x^{3} - 16\,n^{2} x + 4\,n^{2} x^{2} + 6\,n^{2} x^{3} - 32\,n^{3} x - 16\,n^{3} x^{2} + 64\,n^{4} x + 32\,n^{4} x^{2}\right)\,P_{n-2}'(x) + \left(8 - 40\,n^{2} - 12\,n^{2} x + 32\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(-4\,x + 3\,x^{2} + 20\,n^{2} x - 3\,n^{2} x^{2} - 16\,n^{4} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 15 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
215621
3847212168
4420560210201210
5198036002160480308250
69009207901732563009454254411
7400401121121201206160015400168056351008

All 15 rows

Other recurrences

Derivative · proved equal for all n (proof step)

\(\left(-2\,n + n^{2} + n^{3}\right)\,P_{n}(x) = \left(-2\,x + 2\,n - 3\,n x + 6\,n^{2} + 2\,n^{2} x + 4\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(-x - x^{2} - 3\,n x - 3\,n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
Technical details
Node id
n-4b52bcea0ddc9e78 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:derivative (equal by e-45b391e15b1cf3a9)
Fingerprint
f6-63d66b489de6a076

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      2;
  2:     15,       6;
  3:     84,      72,      12;
  4:    420,     560,     210,      20;
  5:   1980,    3600,    2160,     480,      30;
  6:   9009,   20790,   17325,    6300,     945,     42;
  7:  40040,  112112,  120120,   61600,   15400,   1680,    56;
  8: 175032,  576576,  756756,  504504,  180180,  33264,  2772,   72;
  9: 755820, 2864160, 4455360, 3669120, 1719900, 458640, 65520, 4320, 90;
%F Formula
The row polynomials satisfy (4*n^4+2*n^3-10*n^2+4*n)*P(n,x) = -((48*n^4-48*n^3-12*n^2+12*n)*x^2-(4*n^4-4*n^3-10*n^2+10*n)*x^3)*P(n-3,x) - ((16*n^4-16*n^3-4*n^2+4*n)*x^3+(3*n^2-3*n)*x^4)*d/dx P(n-3,x) - (64*n^4-32*n^3-16*n^2+8*n-(16*n^4-8*n^3-16*n^2+8*n)*x+(12*n^4-6*n^3-12*n^2+6*n)*x^2)*P(n-2,x) + ((64*n^4-32*n^3-16*n^2+8*n)*x+(32*n^4-16*n^3+4*n^2-2*n)*x^2+(6*n^2-3*n)*x^3)*d/dx P(n-2,x) + (32*n^4-40*n^2+8+(12*n^4-12*n^2)*x)*P(n-1,x) - ((16*n^4-20*n^2+4)*x+(3*n^2-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 15 + 6*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 15 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((48*n^4 - 48*n^3 - 12*n^2 + 12*n)*x^2 - (4*n^4 - 4*n^3 - 10*n^2 + 10*n)*x^3)*P[n - 3, x] - ((16*n^4 - 16*n^3 - 4*n^2 + 4*n)*x^3 + (3*n^2 - 3*n)*x^4)*D[P[n - 3, x], x] - (64*n^4 - 32*n^3 - 16*n^2 + 8*n - (16*n^4 - 8*n^3 - 16*n^2 + 8*n)*x + (12*n^4 - 6*n^3 - 12*n^2 + 6*n)*x^2)*P[n - 2, x] + ((64*n^4 - 32*n^3 - 16*n^2 + 8*n)*x + (32*n^4 - 16*n^3 + 4*n^2 - 2*n)*x^2 + (6*n^2 - 3*n)*x^3)*D[P[n - 2, x], x] + (32*n^4 - 40*n^2 + 8 + (12*n^4 - 12*n^2)*x)*P[n - 1, x] - ((16*n^4 - 20*n^2 + 4)*x + (3*n^2 - 3)*x^2)*D[P[n - 1, x], x])/(4*n^4 + 2*n^3 - 10*n^2 + 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 15 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = (-((48*n^4 - 48*n^3 - 12*n^2 + 12*n)*x^2 - (4*n^4 - 4*n^3 - 10*n^2 + 10*n)*x^3)*v[i - 3] - ((16*n^4 - 16*n^3 - 4*n^2 + 4*n)*x^3 + (3*n^2 - 3*n)*x^4)*deriv(v[i - 3], x) - (64*n^4 - 32*n^3 - 16*n^2 + 8*n - (16*n^4 - 8*n^3 - 16*n^2 + 8*n)*x + (12*n^4 - 6*n^3 - 12*n^2 + 6*n)*x^2)*v[i - 2] + ((64*n^4 - 32*n^3 - 16*n^2 + 8*n)*x + (32*n^4 - 16*n^3 + 4*n^2 - 2*n)*x^2 + (6*n^2 - 3*n)*x^3)*deriv(v[i - 2], x) + (32*n^4 - 40*n^2 + 8 + (12*n^4 - 12*n^2)*x)*v[i - 1] - ((16*n^4 - 20*n^2 + 4)*x + (3*n^2 - 3)*x^2)*deriv(v[i - 1], x))/(4*n^4 + 2*n^3 - 10*n^2 + 4*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