A199478 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-2}(x) + \left(2\,x\right)\,P_{n-2}'(x) + \left(1\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2 + x\)

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	1								3
1	4	3								7
2	8	11	2							21
3	16	31	8							55
4	32	91	42	4						169
5	64	247	152	20						483
6	128	675	586	136	8					1533
7	256	1791	1992	600	48					4687
8	512	4747	6858	2860	400	16				15393
9	1024	12423	22392	11644	2080	112				49675
10	2048	32435	73034	48240	11800	1104	32			168693
11	4096	84175	232232	186176	55888	6608	256			569431
12	8192	218011	735306	718164	270368	43456	2912	64		1996473
13	16384	562903	2297048	2672036	1201600	234528	19712	576		7004787
14	32768	1451331	7144906	9887960	5341608	1296736	147392	7424	128	25310253

Technical details

Node id: n-811ccbfd604dd9f0 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-921c6d767cd7fcd1

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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:    2,     1;
  1:    4,     3;
  2:    8,    11,     2;
  3:   16,    31,     8;
  4:   32,    91,    42,     4;
  5:   64,   247,   152,    20;
  6:  128,   675,   586,   136,    8;
  7:  256,  1791,  1992,   600,   48;
  8:  512,  4747,  6858,  2860,  400,  16;
  9: 1024, 12423, 22392, 11644, 2080, 112;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + x;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], 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] = 2 + x; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.?
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741