A113413 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 6 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	6														12
2	34	72	36													142
3	194	624	648	216												1682
4	1106	4776	7560	5184	1296											19922
5	6306	34176	72792	76032	38880	7776										235962
6	35954	234456	627912	883008	686880	279936	46656									2794802
7	204994	1562544	5042520	8923392	9344160	5785344	1959552	279936								33102442
8	1168786	10196232	38506248	82171584	108306720	90232704	46376064	13436928	1679616							392074882
9	6663906	65474016	283253976	707844096	1125480096	1180210176	815826816	358318080	90699264	10077696						4643848122
10	37994674	415151160	2024316360	5798468160	10800721440	13664702592	11887295616	7018555392	2690744832	604661760	60466176					55003078162
11	216628994	2605603344	14138348760	45669139200	97543401120	144601562880	151769261952	112744783872	58077762048	19752284160	3990767616	362797056				651472341002
12	1235123666	16216363368	96913536264	348515559360	839725535520	1427731360128	1755993085056	1573737486336	1019731825152	465791109120	142337378304	26121388032	2176782336			7716226532642
13	7042134306	100214746176	654052876632	2591565663744	6953909473440	13340676672000	18821040011136	19766240649216	15449863795200	8874822205440	3640910321664	1010027003904	169789022208	13060694016		91393215269082
14	40151166194	615602732184	4356505185480	18857689033536	55768936847904	119176118854272	189739620588672	228588351768576	209368115089920	145054849185792	74817581008896	27854106771456	7074542592000	1097098297344	78364164096	1082487633286322

Technical details

Node id: n-450cfba591249146 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-65e7907e2a8fdd0f

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:       6,        6;
  2:      34,       72,        36;
  3:     194,      624,       648,       216;
  4:    1106,     4776,      7560,      5184,       1296;
  5:    6306,    34176,     72792,     76032,      38880,       7776;
  6:   35954,   234456,    627912,    883008,     686880,     279936,     46656;
  7:  204994,  1562544,   5042520,   8923392,    9344160,    5785344,   1959552,    279936;
  8: 1168786, 10196232,  38506248,  82171584,  108306720,   90232704,  46376064,  13436928,  1679616;
  9: 6663906, 65474016, 283253976, 707844096, 1125480096, 1180210176, 815826816, 358318080, 90699264, 10077696;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[(4 + 6*x)*P[n - 2, x] + (5 + 6*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] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 6*x)*v[i - 2] + (5 + 6*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_A013610 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)

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