A261711 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-32\,n - 4\,n x + 8\,n x^{2} + 18\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-6 + 8\,x + 2\,x^{2} - 4\,x^{3} - 121\,n - 29\,n x + 26\,n x^{2} + 4\,n x^{3} + 72\,n^{2} + 36\,n^{2} x + 4\,n^{2} x^{2} + 7\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-60 + 76\,x + 20\,x^{2} - 28\,x^{3} - 8\,x^{4} + 66\,n - 20\,n x - 38\,n x^{2} - 8\,n x^{3} + 6\,n^{2} - 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	4				4
2	16	1			17
3	64	8			72
4	255	48	1		304
5	1012	256	12		1280
6	4002	1277	96	1	5376
7	15776	6096	640	16	22528

All 15 rows

Technical details

Node id: n-f353b9195c036490 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-febf4ca7d5f47465

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:      4;
  2:     16,      1;
  3:     64,      8;
  4:    255,     48,     1;
  5:   1012,    256,    12;
  6:   4002,   1277,    96,    1;
  7:  15776,   6096,   640,   16;
  8:  62013,  28200,  3834,  160,  1;
  9: 243140, 127392, 21384, 1280, 20;

%F Formula

The row polynomials satisfy (2*n^3+18*n^2-32*n+(8*n^2-4*n)*x+8*n*x^2)*P(n,x) = (7*n^3+72*n^2-121*n-6+(n^3+36*n^2-29*n+8)*x+(4*n^2+26*n+2)*x^2+(4*n-4)*x^3)*P(n-1,x) + (6*n^2+66*n-60-(4*n^2+20*n-76)*x-(2*n^2+38*n-20)*x^2-(8*n+28)*x^3-8*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[((7*n^3 + 72*n^2 - 121*n - 6 + (n^3 + 36*n^2 - 29*n + 8)*x + (4*n^2 + 26*n + 2)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (6*n^2 + 66*n - 60 - (4*n^2 + 20*n - 76)*x - (2*n^2 + 38*n - 20)*x^2 - (8*n + 28)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 18*n^2 - 32*n + (8*n^2 - 4*n)*x + 8*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] = ((7*n^3 + 72*n^2 - 121*n - 6 + (n^3 + 36*n^2 - 29*n + 8)*x + (4*n^2 + 26*n + 2)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (6*n^2 + 66*n - 60 - (4*n^2 + 20*n - 76)*x - (2*n^2 + 38*n - 20)*x^2 - (8*n + 28)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 18*n^2 - 32*n + (8*n^2 - 4*n)*x + 8*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 read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string…?
  - 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.