A085478 → D → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-12 + 12\,x + 14\,n - 32\,n x - 4\,n^{2} + 31\,n^{2} x - 13\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(6\,n - 2\,n x - n x^{2} - 4\,n^{2} + 9\,n^{2} x + 3\,n^{2} x^{2} - 9\,n^{3} x - 3\,n^{3} x^{2} + 2\,n^{4} x + n^{4} x^{2}\right)\,P_{n-1}(x) + \left(8\,x - 2\,x^{2} - x^{3} - 8\,n x + 10\,n x^{2} + 3\,n x^{3} - 12\,n^{2} x^{2} - 3\,n^{2} x^{3} + 4\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 0\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	P_n(1)
0							0
1							0
2	1						1
3	3	4					7
4	6	20	9				35
5	10	60	63	16			149
6	15	140	252	144	25		576
7	21	280	756	720	275	36	2088

All 15 rows

Technical details

Node id: n-e61fba455c72f1ad · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b29d52f399e218be

Observed on the stored rows ?

Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
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:  0;
  2:  1;
  3:  3,   4;
  4:  6,  20,    9;
  5: 10,  60,   63,   16;
  6: 15, 140,  252,  144,   25;
  7: 21, 280,  756,  720,  275,   36;
  8: 28, 504, 1890, 2640, 1650,  468,  49;
  9: 36, 840, 4158, 7920, 7150, 3276, 735, 64;

%F Formula

The row polynomials satisfy -(4*n^2-14*n+12-(2*n^4-13*n^3+31*n^2-32*n+12)*x)*P(n,x) = -(4*n^2-6*n-(2*n^4-9*n^3+9*n^2-2*n)*x-(n^4-3*n^3+3*n^2-n)*x^2)*P(n-1,x) - ((8*n-8)*x-(4*n^3-12*n^2+10*n-2)*x^2-(n^3-3*n^2+3*n-1)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 6*n - (2*n^4 - 9*n^3 + 9*n^2 - 2*n)*x - (n^4 - 3*n^3 + 3*n^2 - n)*x^2)*P[n - 1, x] - ((8*n - 8)*x - (4*n^3 - 12*n^2 + 10*n - 2)*x^2 - (n^3 - 3*n^2 + 3*n - 1)*x^3)*D[P[n - 1, x], x])/(-4*n^2 + 14*n - 12 + (2*n^4 - 13*n^3 + 31*n^2 - 32*n + 12)*x)]];
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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 6*n - (2*n^4 - 9*n^3 + 9*n^2 - 2*n)*x - (n^4 - 3*n^3 + 3*n^2 - n)*x^2)*v[i - 1] - ((8*n - 8)*x - (4*n^3 - 12*n^2 + 10*n - 2)*x^2 - (n^3 - 3*n^2 + 3*n - 1)*x^3)*deriv(v[i - 1], x))/(-4*n^2 + 14*n - 12 + (2*n^4 - 13*n^3 + 31*n^2 - 32*n + 12)*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA …?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023