A131222 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2} + 6\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(1 - 3\,x + 3\,n x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	1	3						4
3	1	9	14					24
4	1	18	83	90				192
5	1	30	275	870	744			1920
6	1	45	685	4275	10474	7560		23040
7	1	63	1435	14805	70924	143892	91440	322560

All 15 rows

Technical details

Node id: n-e395a1d44939f46e · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-bd19e0a95c26bd62

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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: 1;
  2: 1,   3;
  3: 1,   9,   14;
  4: 1,  18,   83,    90;
  5: 1,  30,  275,   870,     744;
  6: 1,  45,  685,  4275,   10474,    7560;
  7: 1,  63, 1435, 14805,   70924,  143892,    91440;
  8: 1,  84, 2674, 41160,  324289, 1274196,  2233356,  1285200;
  9: 1, 108, 4578, 98280, 1151409, 7398972, 24870572, 38769840, 20603520;

%F Formula

The row polynomials satisfy P(n,x) = -(2*n^2-6*n+4)*x^2*P(n-2,x) + (1+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4)*x^2*P[n - 2, x] + (1 + (3*n - 3)*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] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 6*n + 4)*x^2*v[i - 2] + (1 + (3*n - 3)*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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