A110552 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-8 - 32\,x - 32\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	8														11
2	10	52	64													126
3	36	272	656	512												1476
4	136	1328	4704	7232	4096											17496
5	528	6272	29056	66048	73984	32768										208656
6	2080	28992	165120	494080	821760	721920	262144									2496096
7	8256	131840	888576	3287040	7224320	9449472	6819840	2097152								29906496
8	32896	591616	4598272	20220928	55121920	95506432	102817792	62930944	16777216							358598016
9	131328	2625536	23097344	117555200	381968384	822542336	1174863872	1074003968	570490880	134217728						4301496576
10	524800	11543552	113319936	654655488	2467282944	6343852032	11277041664	13692567552	10872815616	5100535808	1073741824					51607881216
11	2098176	50352128	545443840	3524198400	15102935040	45105414144	95845220352	144970874880	153020006400	107379425280	45098205184	8589934592				619234108416
12	8390656	218148864	2584141824	18457640960	88594513920	301214859264	744163639296	1346558754816	1771760517120	1653620080640	1039405154304	395141185536	68719476736			7430446503936
13	33558528	939622400	12080676864	94496489472	502006743040	1913511739392	5386131210240	11339128897536	17858993061888	20788103086080	17386304438272	9895705313280	3435990614016	549755813888		89163181264896
14	134225920	4026744832	55837130752	474612629504	2763905171456	11669763522560	36851719077888	88443765915648	162146304393216	226021357912064	235847643299840	178671948136448	92909168754688	29686881058816	4398046511104	1069945114484736

Technical details

Node id: n-6ed12f7325828d66 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e48c611829f57832

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      3,       8;
  2:     10,      52,       64;
  3:     36,     272,      656,       512;
  4:    136,    1328,     4704,      7232,      4096;
  5:    528,    6272,    29056,     66048,     73984,     32768;
  6:   2080,   28992,   165120,    494080,    821760,    721920,     262144;
  7:   8256,  131840,   888576,   3287040,   7224320,   9449472,    6819840,    2097152;
  8:  32896,  591616,  4598272,  20220928,  55121920,  95506432,  102817792,   62930944,  16777216;
  9: 131328, 2625536, 23097344, 117555200, 381968384, 822542336, 1174863872, 1074003968, 570490880, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = -(8+32*x+32*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 32*x + 32*x^2)*P[n - 2, x] + (6 + 12*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] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 32*x + 32*x^2)*v[i - 2] + (6 + 12*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 A triangular array related to A077028 and distributing the values of A007582.?
  - 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