A122542 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 18\,x - 18\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 6\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 4 + 3\,x\)

\(P_{3}(x) = 15 + 30\,x + 9\,x^{2}\)

\(P_{4}(x) = 46 + 165\,x + 144\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	4	3							7
3	15	30	9						54
4	46	165	144	27					382
5	137	696	1098	594	81				2606
6	392	2583	5976	5832	2268	243			17294
7	1095	8862	27099	39150	27297	8262	729		112494
8	3002	28809	109656	211383	218538	117855	29160	2187	720590

All 15 rows

Technical details

Node id: n-ba338b6b81ab1684 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0569c33d152ea32b

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,      3;
   3:    15,     30,       9;
   4:    46,    165,     144,      27;
   5:   137,    696,    1098,     594,      81;
   6:   392,   2583,    5976,    5832,    2268,     243;
   7:  1095,   8862,   27099,   39150,   27297,    8262,     729;
   8:  3002,  28809,  109656,  211383,  218538,  117855,   29160,    2187;
   9:  8113,  90048,  410292,  991980, 1373922, 1097388,  481140,  100602,   6561;
  10: 21676, 273147, 1449648, 4220856, 7362576, 7888266, 5111748, 1885194, 341172, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-4,x) - (4+18*x+18*x^2)*P(n-3,x) - (2+6*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4 + 3*x, P(3,x) = 15 + 30*x + 9*x^2, P(4,x) = 46 + 165*x + 144*x^2 + 27*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[3, x_] := 15 + 30*x + 9*x^2;
P[4, x_] := 46 + 165*x + 144*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 4, x] - (4 + 18*x + 18*x^2)*P[n - 3, x] - (2 + 6*x + 9*x^2)*P[n - 2, x] + (4 + 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 + 3*x; v[3] = 15 + 30*x + 9*x^2; v[4] = 46 + 165*x + 144*x^2 + 27*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 6*x + 9*x^2)*v[i - 4] - (4 + 18*x + 18*x^2)*v[i - 3] - (2 + 6*x + 9*x^2)*v[i - 2] + (4 + 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 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_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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.