A350512 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 32\,x + 32\,x^{2} - 4\,n - 16\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 16\,x + 4\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 6 + 24\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	6	24	16													46
3	16	96	160	64												336
4	40	320	864	896	256											2376
5	96	960	3584	6144	4608	1024										16416
6	224	2688	12800	30720	38400	22528	4096									111456
7	512	7168	41472	128000	225280	221184	106496	16384								746496
8	1152	18432	125440	473088	1075200	1490944	1204224	491520	65536							4945536
9	2560	46080	360448	1605632	4472832	8028160	9175040	6291456	2228224	262144						32472576
10	5632	112640	995328	5111808	16859136	37158912	55050240	53477376	31850496	9961472	1048576					211631616
11	12288	270336	2662400	15482880	58982400	154140672	280756224	353894400	298844160	157286400	44040192	4194304				1370566656
12	26624	638976	6938624	45056000	194641920	588251136	1271660544	1972371456	2162688000	1614807040	761266176	192937984	16777216			8828061696
13	57344	1490944	17694720	126877696	612761600	2102132736	5259657216	9688842240	13079937024	12687769600	8489271296	3623878656	838860800	67108864		56596340736
14	122880	3440640	44302336	347602944	1855586304	7122452480	20242759680	43184553984	69275222016	82770395136	71974256640	43620761600	17012097024	3623878656	268435456	361345867776

Technical details

Node id: n-25833e4dbf2c97ec · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-93aff0ca244bb8a2

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,     4;
  2:    6,    24,     16;
  3:   16,    96,    160,      64;
  4:   40,   320,    864,     896,     256;
  5:   96,   960,   3584,    6144,    4608,    1024;
  6:  224,  2688,  12800,   30720,   38400,   22528,    4096;
  7:  512,  7168,  41472,  128000,  225280,  221184,  106496,   16384;
  8: 1152, 18432, 125440,  473088, 1075200, 1490944, 1204224,  491520,   65536;
  9: 2560, 46080, 360448, 1605632, 4472832, 8028160, 9175040, 6291456, 2228224, 262144;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(16*n-32)*x+(16*n-32)*x^2)*P(n-2,x) + (4*n-8+(8*n-16)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 24*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 6 + 24*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (16*n - 32)*x + (16*n - 32)*x^2)*P[n - 2, x] + (4*n - 8 + (8*n - 16)*x)*P[n - 1, x])/(n - 2)]];
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 + 4*x; v[3] = 6 + 24*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (16*n - 32)*x + (16*n - 32)*x^2)*v[i - 2] + (4*n - 8 + (8*n - 16)*x)*v[i - 1])/(n - 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 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.?
  - 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