A372873 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 28\,x - 64\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 16\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

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	8														10
2	5	36	64													105
3	14	144	480	512												1150
4	41	552	2704	5632	4096											13025
5	122	2040	13280	41600	61440	32768										151250
6	365	7308	59568	250944	568320	638976	262144									1787625
7	1094	25536	250432	1329664	4089344	7168000	6422528	2097152								21383750
8	3281	87504	1003616	6437120	25088256	60211200	85327872	62914560	16777216							257850625
9	9842	295272	3878208	29170944	137820672	421152768	823001088	972029952	603979776	134217728						3125556250
10	29525	984180	14566240	125722240	698282240	2594661376	6482903040	10624696320	10695475200	5704253440	1073741824					38015315625
11	88574	3247728	53479712	520948736	3326231040	14561783808	44364390400	93311336448	131076194304	114420613120	53150220288	8589934592				463378468750
12	265721	10628856	192737136	2091461376	15092795136	76084721664	273656811520	703621890048	1272802443264	1558334930944	1195879956480	489626271744	68719476736			5656114390625
13	797162	34543704	683965984	8181506944	65846603264	375663069184	1558280888320	4738905440256	10503893024768	16612858003456	17964774457344	12255689179136	4466765987840	549755813888		69101333281250
14	2391485	111602652	2395737680	31318212800	278116043520	1771491972096	8326451130368	29245437394944	76804202954752	149168446242816	208984451579904	201772194856960	123506079563776	40407052320768	4398046511104	844695798515625

Technical details

Node id: n-390f0d4f48e5b502 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-614c22d6d15f5d3a

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,      8;
  2:    5,     36,      64;
  3:   14,    144,     480,      512;
  4:   41,    552,    2704,     5632,      4096;
  5:  122,   2040,   13280,    41600,     61440,     32768;
  6:  365,   7308,   59568,   250944,    568320,    638976,    262144;
  7: 1094,  25536,  250432,  1329664,   4089344,   7168000,   6422528,   2097152;
  8: 3281,  87504, 1003616,  6437120,  25088256,  60211200,  85327872,  62914560,  16777216;
  9: 9842, 295272, 3878208, 29170944, 137820672, 421152768, 823001088, 972029952, 603979776, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = -(3+28*x+64*x^2)*P(n-2,x) + (4+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 28*x + 64*x^2)*P[n - 2, x] + (4 + 16*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 28*x + 64*x^2)*v[i - 2] + (4 + 16*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 flattened Catalan words of length n with exactly k runs of descents.?
  - 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