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	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	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
8	28	504	1890	2640	1650	468	49							7229
9	36	840	4158	7920	7150	3276	735	64						24179
10	45	1320	8316	20592	25025	16380	5880	1088	81					78727
11	55	1980	15444	48048	75075	65520	33320	9792	1539	100				250873
12	66	2860	27027	102960	200200	222768	149940	62016	15390	2100	121			785448
13	78	4004	45045	205920	486200	668304	569772	310080	107730	23100	2783	144		2423160
14	91	5460	72072	388960	1093950	1813968	1899240	1302336	592515	177100	33396	3600	169	7382857

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