A056241 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 12\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 8\,x\right)\,P_{n-1}(x)\)

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

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

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	5	20	16													41
3	14	84	144	64												306
4	41	328	880	896	256											2401
5	122	1220	4480	7360	5120	1024										19326
6	365	4380	20448	46784	53760	27648	4096									157481
7	1094	15316	86800	255360	413952	358400	143360	16384								1290666
8	3281	52496	349984	1260032	2648832	3268608	2236416	720896	65536							10606081
9	9842	177156	1358208	5787264	14980608	24136704	23740416	13271040	3538944	262144						87262326
10	29525	590500	5117680	25194880	77553920	154639360	199741440	161710080	75694080	17039360	1048576					718359401
11	88574	1948628	18836752	105226176	375654400	894215168	1435529216	1533886464	1046740992	418119680	80740352	4194304				5915180706
12	265721	6377304	68024592	425153920	1727543808	4784412672	9200078848	12263522304	11094589440	6499598336	2249195520	377487360	16777216			48713027041
13	797162	20726212	241805824	1671914816	7620341248	24083884032	54027976704	86519267328	97971208192	76397674496	38986055680	11832131584	1744830464	67108864		401185722606
14	2391485	66961580	848180032	6428312128	32483811584	115412039680	296129089536	554558767104	756391673856	740510138368	504869421056	227100590080	61069066240	7985954816	268435456	3304124833001

Technical details

Node id: n-3c823179bb708635 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-c913de3b43fe0d44

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:    5,     20,      16;
  3:   14,     84,     144,      64;
  4:   41,    328,     880,     896,      256;
  5:  122,   1220,    4480,    7360,     5120,     1024;
  6:  365,   4380,   20448,   46784,    53760,    27648,     4096;
  7: 1094,  15316,   86800,  255360,   413952,   358400,   143360,    16384;
  8: 3281,  52496,  349984, 1260032,  2648832,  3268608,  2236416,   720896,   65536;
  9: 9842, 177156, 1358208, 5787264, 14980608, 24136704, 23740416, 13271040, 3538944, 262144;

%F Formula

The row polynomials satisfy P(n,x) = -(3+12*x+16*x^2)*P(n-2,x) + (4+8*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[-(3 + 12*x + 16*x^2)*P[n - 2, x] + (4 + 8*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] = -(3 + 12*x + 16*x^2)*v[i - 2] + (4 + 8*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 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

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