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	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

All 15 rows

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