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	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

All 15 rows

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.