A008556 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 6\,n x + 4\,n x^{2} + n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(12 + 16\,x - 4\,x^{2} - 8\,x^{3} - 26\,n - 38\,n x + 4\,n x^{2} + 8\,n x^{3} + 6\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 24\,x^{3} - 16\,x^{4} - 12\,n x - 28\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	8	2			10
3	32	12			44
4	130	72	6		208
5	532	400	60		992
6	2184	2100	480	20	4784
7	8976	10584	3360	280	23200

All 15 rows

Technical details

Node id: n-391a382dd61abb01 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c4beaef4f26d1be9

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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:      2;
  2:      8,      2;
  3:     32,     12;
  4:    130,     72,      6;
  5:    532,    400,     60;
  6:   2184,   2100,    480,    20;
  7:   8976,  10584,   3360,   280;
  8:  36894,  51744,  21420,  2800,   70;
  9: 151580, 247104, 127512, 23520, 1260;

%F Formula

The row polynomials satisfy (n^3+n^2-4*n+(4*n^2-6*n)*x+4*n*x^2)*P(n,x) = (4*n^3+6*n^2-26*n+12+(2*n^3+16*n^2-38*n+16)*x+(8*n^2+4*n-4)*x^2+(8*n-8)*x^3)*P(n-1,x) - ((4*n^2+12*n-8)*x+(4*n^2+28*n)*x^2+(16*n+24)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 6*n^2 - 26*n + 12 + (2*n^3 + 16*n^2 - 38*n + 16)*x + (8*n^2 + 4*n - 4)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - ((4*n^2 + 12*n - 8)*x + (4*n^2 + 28*n)*x^2 + (16*n + 24)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 - 4*n + (4*n^2 - 6*n)*x + 4*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 6*n^2 - 26*n + 12 + (2*n^3 + 16*n^2 - 38*n + 16)*x + (8*n^2 + 4*n - 4)*x^2 + (8*n - 8)*x^3)*v[i - 1] - ((4*n^2 + 12*n - 8)*x + (4*n^2 + 28*n)*x^2 + (16*n + 24)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 - 4*n + (4*n^2 - 6*n)*x + 4*n*x^2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of coefficients of Legendre polynomials 2^n P_n (x).?
  - 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 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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.