A122542 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 4\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 12\,x - 8\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 4\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 2\)

\(P_{3}(x) = 8 + 8\,x\)

\(P_{4}(x) = 26 + 56\,x + 24\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	2														2
3	8	8													16
4	26	56	24												106
5	80	256	240	64											640
6	234	992	1440	832	160										3658
7	664	3496	6864	6208	2560	384									20176
8	1842	11576	28632	35392	22880	7296	896								108514
9	5024	36672	109344	171648	152320	76032	19712	2048							572800
10	13522	112384	392064	746624	845120	579840	234752	51200	4608						2980114
11	36008	335624	1340640	3001984	4138240	3629568	2019584	686080	129024	10240					15326992
12	95050	981752	4417368	11371136	18493120	19799040	14095872	6576128	1921536	317440	22528				78090970
13	249072	2823360	14127504	41098944	77057280	97542912	84878080	50673664	20312064	5201920	765952	49152			394779904
14	648634	8004960	44087712	143022016	303782880	444205824	457649920	334149632	171357696	60129280	13697024	1818624	106496		1982660698
15	1680184	22424104	134787888	482383552	1144925440	1899970944	2264258304	1959036928	1228879872	551393280	171888640	35241984	4259840	229376	9901360336

Technical details

Node id: n-a7c181b3c3a9e0c6 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5d68b0a72c589ca1

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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:     0;
   2:     2;
   3:     8,      8;
   4:    26,     56,     24;
   5:    80,    256,    240,     64;
   6:   234,    992,   1440,    832,    160;
   7:   664,   3496,   6864,   6208,   2560,    384;
   8:  1842,  11576,  28632,  35392,  22880,   7296,    896;
   9:  5024,  36672, 109344, 171648, 152320,  76032,  19712,  2048;
  10: 13522, 112384, 392064, 746624, 845120, 579840, 234752, 51200, 4608;

%F Formula

The row polynomials satisfy P(n,x) = -(1+4*x+4*x^2)*P(n-4,x) - (4+12*x+8*x^2)*P(n-3,x) - (2+4*x+4*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 8 + 8*x, P(4,x) = 26 + 56*x + 24*x^2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 8 + 8*x;
P[4, x_] := 26 + 56*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 4*x^2)*P[n - 4, x] - (4 + 12*x + 8*x^2)*P[n - 3, x] - (2 + 4*x + 4*x^2)*P[n - 2, x] + (4 + 4*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] = 0; v[2] = 2; v[3] = 8 + 8*x; v[4] = 26 + 56*x + 24*x^2; for(n = 5, N, my(i = n); v[i] = -(1 + 4*x + 4*x^2)*v[i - 4] - (4 + 12*x + 8*x^2)*v[i - 3] - (2 + 4*x + 4*x^2)*v[i - 2] + (4 + 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, 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_A013609 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.