A146986 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(24 + 48\,x + 24\,x^{2} - 8\,n - 16\,n x - 8\,n x^{2}\right)\,P_{n-2}(x) + \left(-18 - 18\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 6 + 12\,x + 4\,x^{2}\)

\(P_{3}(x) = 16 + 48\,x + 40\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	6	12	4						22
3	16	48	40	8					112
4	48	192	256	128	16				640
5	160	800	1472	1216	416	32			4096
6	576	3456	8128	9472	5568	1408	64		28672
7	2176	15232	43648	65920	55680	25216	4992	128	212992

All 15 rows

Technical details

Node id: n-e34f9664814ccfa0 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9e2859b846aec920

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:     2,      2;
  2:     6,     12,       4;
  3:    16,     48,      40,       8;
  4:    48,    192,     256,     128,      16;
  5:   160,    800,    1472,    1216,     416,      32;
  6:   576,   3456,    8128,    9472,    5568,    1408,      64;
  7:  2176,  15232,   43648,   65920,   55680,   25216,    4992,    128;
  8:  8448,  67584,  228352,  423936,  468480,  309248,  113664,  18432,   256;
  9: 33280, 299520, 1165312, 2566144, 3505152, 3046400, 1648640, 509952, 70144, 512;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(16*n-48)*x+(8*n-24)*x^2)*P(n-2,x) + (6*n-18+(6*n-18)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 6 + 12*x + 4*x^2, P(3,x) = 16 + 48*x + 40*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 6 + 12*x + 4*x^2;
P[3, x_] := 16 + 48*x + 40*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (16*n - 48)*x + (8*n - 24)*x^2)*P[n - 2, x] + (6*n - 18 + (6*n - 18)*x)*P[n - 1, x])/(n - 3)]];
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] = 2 + 2*x; v[3] = 6 + 12*x + 4*x^2; v[4] = 16 + 48*x + 40*x^2 + 8*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (16*n - 48)*x + (8*n - 24)*x^2)*v[i - 2] + (6*n - 18 + (6*n - 18)*x)*v[i - 1])/(n - 3)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.?
  - 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