A119900 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

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	4	14	12													30
3	8	40	64	32												144
4	16	104	244	240	80											684
5	32	256	792	1168	800	192										3240
6	64	608	2336	4600	4816	2464	448									15336
7	128	1408	6464	15936	22528	17920	7168	1024								72576
8	256	3200	17088	50624	90256	97920	61824	19968	2304							343440
9	512	7168	43648	151040	325280	448064	389376	201216	53760	5120						1625184
10	1024	15872	108544	429696	1085120	1814240	2014144	1444608	625152	140800	11264					7690464
11	2048	34816	264192	1177600	3412352	6714240	9088000	8374784	5068800	1869824	360448	24576				36391680
12	4096	75776	631808	3131392	10241536	23188864	37064000	41816320	32690944	16986112	5417984	905216	53248			172207296
13	8192	163840	1488896	8122368	29596672	75808768	139805568	186855680	179545600	121099264	54765568	15282176	2236416	114688		814893696
14	16384	352256	3465216	20633600	82892800	236995584	495502336	766050176	874625280	727733760	429192192	170846208	42119168	5447680	245760	3856118400

Technical details

Node id: n-3005506063937011 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9b02f85dc5a0184c

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:   4,   14,    12;
  3:   8,   40,    64,     32;
  4:  16,  104,   244,    240,     80;
  5:  32,  256,   792,   1168,    800,    192;
  6:  64,  608,  2336,   4600,   4816,   2464,    448;
  7: 128, 1408,  6464,  15936,  22528,  17920,   7168,   1024;
  8: 256, 3200, 17088,  50624,  90256,  97920,  61824,  19968,  2304;
  9: 512, 7168, 43648, 151040, 325280, 448064, 389376, 201216, 53760, 5120;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2*x + 4*x^2)*P[n - 2, x] + (2 + 4*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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*x + 4*x^2)*v[i - 2] + (2 + 4*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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, 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