A079510 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{2} - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(x^{2} - n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(2\,x + 2\,n x - n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 12\,x\)

\(P_{3}(x) = 40\,x + 72\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	3	12						15
3	0	40	72					112
4	0	30	390	480				900
5	0	0	630	3696	3600			7926
6	0	0	315	9520	36540	30240		76615
7	0	0	0	10080	132160	385344	282240	809824

All 15 rows

Technical details

Node id: n-de13736ed4fe54c9 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9d47e81a561df7f5

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 0;
  1: 2;
  2: 3, 12;
  3: 0, 40,  72;
  4: 0, 30, 390,   480;
  5: 0,  0, 630,  3696,   3600;
  6: 0,  0, 315,  9520,  36540,   30240;
  7: 0,  0,   0, 10080, 132160,  385344,   282240;
  8: 0,  0,   0,  3780, 220500, 1821960,  4363632,  2903040;
  9: 0,  0,   0,     0, 173250, 4231920, 25751880, 53095680, 32659200;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-4,x) - (2*n^2+n-1)*x^2*P(n-3,x) + ((2*n+2)*x-(n^2+n)*x^2)*P(n-2,x) + (2*n+2)*x*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 3 + 12*x, P(3,x) = 40*x + 72*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 3 + 12*x;
P[3, x_] := 40*x + 72*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 4, x] - (2*n^2 + n - 1)*x^2*P[n - 3, x] + ((2*n + 2)*x - (n^2 + n)*x^2)*P[n - 2, x] + (2*n + 2)*x*P[n - 1, 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] = 2; v[3] = 3 + 12*x; v[4] = 40*x + 72*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 4] - (2*n^2 + n - 1)*x^2*v[i - 3] + ((2*n + 2)*x - (n^2 + n)*x^2)*v[i - 2] + (2*n + 2)*x*v[i - 1]); 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 T(n,k) read by rows; related to number of preorders.?
  - 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.