A113413 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	4														11
2	47	56	16													119
3	317	572	336	64												1289
4	2137	5168	4608	1792	256											13961
5	14407	43684	52288	30848	8960	1024										151211
6	97127	354120	532176	421632	185600	43008	4096									1637759
7	654797	2789276	5045712	5021888	2968320	1041408	200704	16384								17738489
8	4414417	21513952	45508736	54551040	40512000	19079168	5562368	917504	65536							192124721
9	29760487	163306948	395493888	554633472	496205312	293603328	114860032	28639232	4128768	262144						2080893611
10	200635007	1224111064	3339790608	5364566528	5616529920	4003885056	1967702016	658112512	143327232	18350080	1048576					22538058599
11	1352612477	9082783100	27565885136	49901704512	59857006080	49942267904	29570465792	12422119424	3627614208	701235200	80740352	4194304				244108628489
12	9118849897	66830243856	223321840512	449855762688	608289770240	581567176704	403045916672	203975950336	74803249152	19382927360	3368026112	352321536	16777216			2643928812281
13	61476161767	488281228132	1781412576576	3952194001280	5948053520640	6411701505024	5092165607424	3016431828992	1332549844992	433527193600	100948508672	15929966592	1526726656	67108864		28636265779211
14	414451220087	3546218634728	14025530549968	33981380489472	56337969030912	67603418077184	60541297954816	41099316822016	21250946433024	8327490174976	2434171666432	514607546368	74373398528	6576668672	268435456	310158017102639

Technical details

Node id: n-eedc1a2f77b2e75f · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-3f270cf733a15daf

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:        7,         4;
  2:       47,        56,        16;
  3:      317,       572,       336,        64;
  4:     2137,      5168,      4608,      1792,       256;
  5:    14407,     43684,     52288,     30848,      8960,      1024;
  6:    97127,    354120,    532176,    421632,    185600,     43008,      4096;
  7:   654797,   2789276,   5045712,   5021888,   2968320,   1041408,    200704,    16384;
  8:  4414417,  21513952,  45508736,  54551040,  40512000,  19079168,   5562368,   917504,   65536;
  9: 29760487, 163306948, 395493888, 554633472, 496205312, 293603328, 114860032, 28639232, 4128768, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (5+4*x)*P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[(5 + 4*x)*P[n - 2, x] + (6 + 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] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 4*x)*v[i - 2] + (6 + 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 basis_map_A013611 of A Riordan array of coordination sequences.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from A Riordan array of coordination sequences.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of A Riordan array of coordination sequences.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A Riordan array of coordination sequences.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A Riordan array of coordination sequences.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from A Riordan array of coordination sequences.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

A113413 → basis A013611 → x ↦ x+1 15 rows; no all-index equality certificate was found

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