A081580 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(25 + 20\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	4														10
2	61	68	16													145
3	516	872	448	64												1900
4	4621	10216	7936	2496	256											25525
5	40626	111900	117120	57280	12800	1024										340750
6	359281	1181724	1553040	1033280	362240	62464	4096									4556125
7	3171336	12137488	19211136	16186240	7772160	2105344	294912	16384								60895000
8	28010041	122238992	226277248	230854784	141099520	52527104	11542528	1359872	65536							813975625
9	247343646	1212338036	2569647616	3079116416	2288045056	1087637504	328843264	60637184	6160384	262144						10880031250
10	2184312901	11879578420	28368948880	39050203520	34189319680	19813173248	7662714880	1944043520	308346880	27525120	1048576					145429215625
11	19289468556	115270045944	306219958080	476147879360	480120186880	328588156928	155202813952	50407915520	10993008640	1528299520	121634816	4194304				1943893562500
12	170344633861	1109453868408	3245215222656	5625401174080	6421049701120	5071125413888	2833400848384	1125114134528	314179256320	59996897280	7419723776	532676608	16777216			25983250328125
13	1504304517066	10605642265612	33870006680448	64761364080704	82553865162240	73928058948608	47736740233216	22448542367744	7668515602432	1874766069760	318112792576	35412508672	2315255808	67108864		347307713593750
14	13284442948921	100794311049356	348962067079696	729587545011136	1027402913305856	1028982943710208	754390206681088	410434085470208	166162027184128	49706166386688	10793172074496	1646637613056	166614532096	9999220736	268435456	4642323400703125

Technical details

Node id: n-4a83e4b905d55ad8 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-31b02e78b9a6fd3c

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,          4;
  2:        61,         68,         16;
  3:       516,        872,        448,         64;
  4:      4621,      10216,       7936,       2496,        256;
  5:     40626,     111900,     117120,      57280,      12800,       1024;
  6:    359281,    1181724,    1553040,    1033280,     362240,      62464,      4096;
  7:   3171336,   12137488,   19211136,   16186240,    7772160,    2105344,    294912,    16384;
  8:  28010041,  122238992,  226277248,  230854784,  141099520,   52527104,  11542528,  1359872,   65536;
  9: 247343646, 1212338036, 2569647616, 3079116416, 2288045056, 1087637504, 328843264, 60637184, 6160384, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (25+20*x)*P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[(25 + 20*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] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (25 + 20*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 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)

Related ?

Candidate equalities ?

A081580 → 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