Hadamard-factor parent of A155495 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n - 5\,n x + 2\,n x^{2} - n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(2\,n - 13\,n x - 7\,n x^{2} + 6\,n x^{3} + 3\,n^{2} + 5\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x + 6\,x^{3} - 4\,x^{4} + 4\,n x + 4\,n x^{2} - 8\,n x^{3} + 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 7 + 8\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	7	8	1								16
2	16	45	18	1							80
3	29	168	154	32	1						384
4	46	465	840	390	50	1					1792
5	67	1056	3267	2904	825	72	1				8192
6	92	2093	10010	15015	8008	1547	98	1			36864
7	121	3760	25844	59488	52910	18928	2660	128	1		163840
8	154	6273	58752	193596	262548	155142	39984	4284	162	1	720896

All 15 rows

Technical details

Node id: n-798f7776a4245470 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4af00dfa4d313d82

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   7,     8,      1;
   2:  16,    45,     18,       1;
   3:  29,   168,    154,      32,       1;
   4:  46,   465,    840,     390,      50,       1;
   5:  67,  1056,   3267,    2904,     825,      72,       1;
   6:  92,  2093,  10010,   15015,    8008,    1547,      98,      1;
   7: 121,  3760,  25844,   59488,   52910,   18928,    2660,    128,      1;
   8: 154,  6273,  58752,  193596,  262548,  155142,   39984,   4284,    162,    1;
   9: 191,  9880, 121125,  542640, 1049750,  940576,  397290,  77520,   6555,  200,   1;
  10: 232, 14861, 231154, 1353693, 3553000, 4526522, 2885036, 916674, 140448, 9625, 242, 1;

%F Formula

The row polynomials satisfy (2*n^3-n^2+n+(4*n^2-5*n)*x+2*n*x^2)*P(n,x) = (2*n^3+3*n^2+2*n+(6*n^3+5*n^2-13*n)*x+(12*n^2-7*n)*x^2+6*n*x^3)*P(n-1,x) + ((4*n^2+4*n-2)*x-(4*n^2-4*n)*x^2-(8*n-6)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 7 + 8*x + x^2.

%t Mathematica

P[1, x_] := 7 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 3*n^2 + 2*n + (6*n^3 + 5*n^2 - 13*n)*x + (12*n^2 - 7*n)*x^2 + 6*n*x^3)*P[n - 1, x] + ((4*n^2 + 4*n - 2)*x - (4*n^2 - 4*n)*x^2 - (8*n - 6)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - n^2 + n + (4*n^2 - 5*n)*x + 2*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 7 + 8*x + x^2; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 3*n^2 + 2*n + (6*n^3 + 5*n^2 - 13*n)*x + (12*n^2 - 7*n)*x^2 + 6*n*x^3)*v[i - 1] + ((4*n^2 + 4*n - 2)*x - (4*n^2 - 4*n)*x^2 - (8*n - 6)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - n^2 + n + (4*n^2 - 5*n)*x + 2*n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
  - 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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

References

[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