A261711 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4								4
2	16	4							20
3	64	32							96
4	256	192	16						464
5	1024	1024	192						2240
6	4096	5120	1536	64					10816
7	16384	24576	10240	1024					52224
8	65536	114688	61440	10240	256				252160
9	262144	524288	344064	81920	5120				1217536
10	1048576	2359296	1835008	573440	61440	1024			5878784
11	4194304	10485760	9437184	3670016	573440	24576			28385280
12	16777216	46137344	47185920	22020096	4587520	344064	4096		137056256
13	67108864	201326592	230686720	125829120	33030144	3670016	114688		661766144
14	268435456	872415232	1107296256	692060160	220200960	33030144	1835008	16384	3195289600

Technical details

Node id: n-1c6e7b2945273445 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-6cf8c5bbd0a3960a

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      4;
  2:     16,      4;
  3:     64,     32;
  4:    256,    192,     16;
  5:   1024,   1024,    192;
  6:   4096,   5120,   1536,    64;
  7:  16384,  24576,  10240,  1024;
  8:  65536, 114688,  61440, 10240,  256;
  9: 262144, 524288, 344064, 81920, 5120;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[4*x*P[n - 2, x] + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = 4*x*v[i - 2] + 4*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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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 words over alphabet {0,1,2,3} having exactly k occurrences of the string…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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