A122542 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 8\,x - 16\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 24\,x - 32\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 8\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 8\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 5 + 4\,x\)

\(P_{3}(x) = 19 + 48\,x + 16\,x^{2}\)

\(P_{4}(x) = 59 + 276\,x + 304\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	4														9
3	19	48	16													83
4	59	276	304	64												703
5	177	1184	2432	1664	256											5713
6	509	4436	13504	17152	8448	1024										45073
7	1427	15312	61904	117632	106752	40960	4096									348083
8	3923	49988	252208	642624	873728	613376	192512	16384								2644743
9	10625	156736	948096	3037952	5560832	5840896	3334144	884736	65536							19839553
10	28437	476580	3361280	12991488	30030336	42518528	36233216	17399808	3997696	262144						147299513
11	75379	1414512	11396368	51606784	144431616	258777088	297222144	212566016	88014848	17825792	1048576					1084379123
12	198219	4117108	37302448	193717824	637264384	1385494528	2015461376	1941241856	1194262528	434372608	78643200	4194304				7926270383
13	517745	11791584	118666752	695256448	2630228736	6735912960	11904851968	14524514304	12021334016	6482296832	2101346304	343932928	16777216			57587427793
14	1344589	33317108	368711616	2405919488	10293140224	30373383168	63317385216	93910761472	98427928576	71320731648	34208743424	9999220736	1493172224	67108864		416220868353
15	3474611	93056784	1123151312	8077571200	38567148288	128925663232	310091419648	542917033984	691875676160	634536329216	408490606592	176353705984	46925873152	6442450944	268435456	2994691596563

Technical details

Node id: n-c8c8dea2b57b05e4 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8e7b21ecb18cceec

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:     5,      4;
   3:    19,     48,      16;
   4:    59,    276,     304,       64;
   5:   177,   1184,    2432,     1664,      256;
   6:   509,   4436,   13504,    17152,     8448,     1024;
   7:  1427,  15312,   61904,   117632,   106752,    40960,     4096;
   8:  3923,  49988,  252208,   642624,   873728,   613376,   192512,    16384;
   9: 10625, 156736,  948096,  3037952,  5560832,  5840896,  3334144,   884736,   65536;
  10: 28437, 476580, 3361280, 12991488, 30030336, 42518528, 36233216, 17399808, 3997696, 262144;

%F Formula

The row polynomials satisfy P(n,x) = -(1+8*x+16*x^2)*P(n-4,x) - (4+24*x+32*x^2)*P(n-3,x) - (2+8*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 5 + 4*x, P(3,x) = 19 + 48*x + 16*x^2, P(4,x) = 59 + 276*x + 304*x^2 + 64*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 4*x;
P[3, x_] := 19 + 48*x + 16*x^2;
P[4, x_] := 59 + 276*x + 304*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 8*x + 16*x^2)*P[n - 4, x] - (4 + 24*x + 32*x^2)*P[n - 3, x] - (2 + 8*x + 16*x^2)*P[n - 2, x] + (4 + 8*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] = 5 + 4*x; v[3] = 19 + 48*x + 16*x^2; v[4] = 59 + 276*x + 304*x^2 + 64*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 8*x + 16*x^2)*v[i - 4] - (4 + 24*x + 32*x^2)*v[i - 3] - (2 + 8*x + 16*x^2)*v[i - 2] + (4 + 8*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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.