A056241 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 6\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)

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

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

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	2														4
2	5	10	4													19
3	14	42	36	8												100
4	41	164	220	112	16											553
5	122	610	1120	920	320	32										3124
6	365	2190	5112	5848	3360	864	64									17803
7	1094	7658	21700	31920	25872	11200	2240	128								101812
8	3281	26248	87496	157504	165552	102144	34944	5632	256							583057
9	9842	88578	339552	723408	936288	754272	370944	103680	13824	512						3340900
10	29525	295250	1279420	3149360	4847120	4832480	3120960	1263360	295680	33280	1024					19147459
11	88574	974314	4709188	13153272	23478400	27944224	22430144	11983488	4088832	816640	78848	2048				109747972
12	265721	3188652	17006148	53144240	107971488	149512896	143751232	95808768	43338240	12694528	2196480	184320	4096			629066809
13	797162	10363106	60451456	208989352	476271328	752621376	844187136	675931776	382700032	149214208	38072320	5777408	425984	8192		3605810836
14	2391485	33480790	212045008	803539016	2030238224	3606626240	4627017024	4332490368	2954654976	1446308864	493036544	110888960	14909440	974848	16384	20668618171

Technical details

Node id: n-2d23a1ca8e96d2ab · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-84dcfd96fb318ee3

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,     2;
  2:    5,    10,      4;
  3:   14,    42,     36,      8;
  4:   41,   164,    220,    112,     16;
  5:  122,   610,   1120,    920,    320,     32;
  6:  365,  2190,   5112,   5848,   3360,    864,     64;
  7: 1094,  7658,  21700,  31920,  25872,  11200,   2240,    128;
  8: 3281, 26248,  87496, 157504, 165552, 102144,  34944,   5632,   256;
  9: 9842, 88578, 339552, 723408, 936288, 754272, 370944, 103680, 13824, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(3+6*x+4*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 6*x + 4*x^2)*P[n - 2, x] + (4 + 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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 6*x + 4*x^2)*v[i - 2] + (4 + 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 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