A131222 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	4	10	4						18
3	24	70	48	8					150
4	192	628	572	176	16				1584
5	1920	6868	7540	3240	560	32			20160
6	23040	88608	111496	59880	14800	1632	64		299520
7	322560	1317552	1843240	1166312	367360	58688	4480	128	5080320

All 15 rows

Technical details

Node id: n-3bc49e97c8fa21a8 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-0c5c3044a5453059

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:        1,         2;
  2:        4,        10,         4;
  3:       24,        70,        48,         8;
  4:      192,       628,       572,       176,        16;
  5:     1920,      6868,      7540,      3240,       560,       32;
  6:    23040,     88608,    111496,     59880,     14800,     1632,      64;
  7:   322560,   1317552,   1843240,   1166312,    367360,    58688,    4480,    128;
  8:  5160960,  22188192,  33820720,  24315424,   9171344,  1888768,  210560,  11776,   256;
  9: 92897280, 417460896, 683451504, 544900096, 236770128, 58988832, 8539776, 701184, 29952, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4)*P[n - 2, x] + (3*n - 2 + 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] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 6*n + 4)*v[i - 2] + (3*n - 2 + 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (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

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