A111006 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 12\,x + 16\,x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	3	20	32													55
3	5	52	176	192												425
4	11	148	736	1600	1280											3775
5	21	356	2384	7872	12800	8192										31625
6	43	868	7232	31808	77824	100352	53248									271375
7	85	2004	20080	110784	363264	707584	757760	344064								2305625
8	171	4596	53664	355392	1459456	3804160	6144000	5619712	2228224							19669375
9	341	10308	137616	1064640	5258240	17188864	37179392	51298304	40960000	14417920						167515625
10	683	22916	344064	3043392	17559040	69029888	187224064	345849856	416350208	294912000	93323264					1427659375
11	1365	50356	840112	8365248	55226624	253776896	828100608	1918517248	3092119552	3301441536	2101346304	603979776				12163765625
12	2731	109844	2015584	22307904	165831424	872145920	3326959616	9273786368	18744672256	26789543936	25694306304	14847836160	3909091328			103648609375
13	5461	237860	4761296	57987776	479341056	2839588864	12400693248	40416296960	98295873536	176172826624	226146385920	196872241152	104169734144	25300041728		883156015625
14	10923	512164	11105728	147593280	1342899200	8848148480	43532021760	162444492800	461967196160	996252057600	1603615588352	1868117835776	1488759816192	726453452800	163745628160	7525238359375

Technical details

Node id: n-bb552e112dcbf0cb · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-84cc0516717215d0

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:   1,     4;
  2:   3,    20,     32;
  3:   5,    52,    176,     192;
  4:  11,   148,    736,    1600,    1280;
  5:  21,   356,   2384,    7872,   12800,     8192;
  6:  43,   868,   7232,   31808,   77824,   100352,    53248;
  7:  85,  2004,  20080,  110784,  363264,   707584,   757760,   344064;
  8: 171,  4596,  53664,  355392, 1459456,  3804160,  6144000,  5619712,  2228224;
  9: 341, 10308, 137616, 1064640, 5258240, 17188864, 37179392, 51298304, 40960000, 14417920;

%F Formula

The row polynomials satisfy P(n,x) = (2+12*x+16*x^2)*P(n-2,x) + (1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 12*x + 16*x^2)*P[n - 2, x] + (1 + 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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 12*x + 16*x^2)*v[i - 2] + (1 + 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)
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 Another version of Fibonacci-Pascal triangle A037027.?
  - 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