A122542 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 4 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	6														10
3	24	60	36													120
4	136	492	576	216												1420
5	776	3660	6336	4752	1296											16820
6	4424	25740	58896	66096	36288	7776										199220
7	25224	174540	496224	740880	611712	264384	46656									2359620
8	143816	1153548	3918384	7299504	8045568	5241024	1866240	279936								27948020
9	819976	7480140	29545344	65948688	90916992	79206336	42550272	12877056	1679616							331024420
10	4675144	47797644	215202384	559723824	926256384	1010771136	726900480	332004096	87340032	10077696						3920748820
11	26655624	301879500	1525860000	4530900240	8748984960	11473721280	10344568320	6328233216	2512705536	584506368	60466176					46438481220
12	151978696	1888572684	10588172400	35339770800	77993694720	119463139008	129537398016	99397995264	52874311680	18563116032	3869835264	362797056				550030781620
13	866515976	11722187340	72187015104	267506649360	664188439680	1163666658240	1474686425088	1361594707200	908779751424	427475708928	134476775424	25395793920	2176782336			6514723410020
14	4940494664	72276195468	484952325264	1975543455024	5449995498240	10749278653632	15590360500992	16830908455680	13521352218624	7981555387392	3364096167936	958509821952	165435457536	13060694016		77162265326420
15	28168537224	443111790540	3217499983584	14290579914768	43365031876224	95086163528640	155328220076544	191991202663680	180396899076096	128198471592960	67812574519296	25885569945600	6739318112256	1070976909312	78364164096	913932152690820

Technical details

Node id: n-23ab3045825b921f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-86c45451eeb81fe1

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       1;
   2:       4,        6;
   3:      24,       60,        36;
   4:     136,      492,       576,       216;
   5:     776,     3660,      6336,      4752,      1296;
   6:    4424,    25740,     58896,     66096,     36288,       7776;
   7:   25224,   174540,    496224,    740880,    611712,     264384,     46656;
   8:  143816,  1153548,   3918384,   7299504,   8045568,    5241024,   1866240,    279936;
   9:  819976,  7480140,  29545344,  65948688,  90916992,   79206336,  42550272,  12877056,  1679616;
  10: 4675144, 47797644, 215202384, 559723824, 926256384, 1010771136, 726900480, 332004096, 87340032, 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 >= 3, with P(1,x) = 1, P(2,x) = 4 + 6*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 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, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 6*x; for(n = 3, N, my(i = n); 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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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