A081580 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(25 + 60\,x\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6 + 12\,x\)

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	12														18
2	61	204	144													409
3	516	2616	4032	1728												8892
4	4621	30648	71424	67392	20736											194821
5	40626	335700	1054080	1546560	1036800	248832										4262598
6	359281	3545172	13977360	27898560	29341440	15178752	2985984									93286549
7	3171336	36412464	172900224	437028480	629544960	511598592	214990848	35831808								2041478712
8	28010041	366716976	2036495232	6233079168	11429061120	12764086272	8414502912	2974040064	429981696							44675973481
9	247343646	3637014108	23126828544	83136143232	185331649536	264295913472	239726739456	132613521408	40418279424	5159780352						977693213178
10	2184312901	35638735260	255320539920	1054355495040	2769334894080	4814601099264	5586119147520	4251623178240	2023063879680	541776936960	61917364224					21395935583089
11	19289468556	345810137832	2755979622720	12855992742720	38889735137280	79846922133504	113142851371008	110242111242240	72125129687040	30081519452160	7182414249984	743008370688				468230763615732
12	170344633861	3328361605224	29206937003904	151885831700160	520105025790720	1232283475574784	2065549218471936	2460624612212736	2061330100715520	1180918929162240	438127269249024	94362063077376	8916100448256			10246808269645741
13	1504304517066	31816926796836	304830060124032	1748556830179008	6686863078141440	17964518324511744	34800083630014464	49094962158256128	50313130867556352	36901020551086080	18784242288820224	6273219673718784	1230421861859328	106993205379072		224242163760960558
14	13284442948921	302382933148068	3140658603717264	19698863715300672	83219635977774336	250042855321580544	549950460670513152	897619344923344896	1090189060355063808	978366472989179904	637326017826914304	291696913240031232	88545793551630336	15941987601481728	1283918464548864	4907337650617178029

Technical details

Node id: n-3c83a3830f60f25e · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-4db3a90e791dd224

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,         12;
  2:        61,        204,         144;
  3:       516,       2616,        4032,        1728;
  4:      4621,      30648,       71424,       67392,        20736;
  5:     40626,     335700,     1054080,     1546560,      1036800,       248832;
  6:    359281,    3545172,    13977360,    27898560,     29341440,     15178752,      2985984;
  7:   3171336,   36412464,   172900224,   437028480,    629544960,    511598592,    214990848,     35831808;
  8:  28010041,  366716976,  2036495232,  6233079168,  11429061120,  12764086272,   8414502912,   2974040064,   429981696;
  9: 247343646, 3637014108, 23126828544, 83136143232, 185331649536, 264295913472, 239726739456, 132613521408, 40418279424, 5159780352;

%F Formula

The row polynomials satisfy P(n,x) = (25+60*x)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[(25 + 60*x)*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] = 6 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (25 + 60*x)*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 basis_map_A013611 of Pascal-(1,5,1) array.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,5,1) array.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    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 Pascal-(1,5,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,5,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,5,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (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