A122542 → D+1 → reverse

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1 + x\)

\(P_{3}(x) = 1 + 4\,x + 2\,x^{2}\)

\(P_{4}(x) = 1 + 7\,x + 10\,x^{2} + 2\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	1	4	2													7
4	1	7	10	2												20
5	1	10	26	18	2											57
6	1	13	50	66	26	2										158
7	1	16	82	166	130	34	2									431
8	1	19	122	338	418	218	42	2								1160
9	1	22	170	602	1042	870	330	50	2							3089
10	1	25	226	978	2202	2610	1586	466	58	2						8154
11	1	28	290	1486	4146	6474	5618	2630	626	66	2					21367
12	1	31	362	2146	7170	14026	16154	10802	4066	810	74	2				55644
13	1	34	442	2978	11618	27470	39930	35546	19058	5958	1018	82	2			144137
14	1	37	530	4002	17882	49762	88034	99330	70906	31442	8370	1250	90	2		371638
15	1	40	626	5238	26402	84722	177506	244878	221954	130826	49170	11366	1506	98	2	954335

Technical details

Node id: n-246d1125f85e1901 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-bf3a8403ddb100bf

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: 1,  1;
   3: 1,  4,   2;
   4: 1,  7,  10,   2;
   5: 1, 10,  26,  18,    2;
   6: 1, 13,  50,  66,   26,    2;
   7: 1, 16,  82, 166,  130,   34,    2;
   8: 1, 19, 122, 338,  418,  218,   42,   2;
   9: 1, 22, 170, 602, 1042,  870,  330,  50,  2;
  10: 1, 25, 226, 978, 2202, 2610, 1586, 466, 58, 2;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*x+2*x^2)*P(n-3,x) - (1+x^2)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 4*x + 2*x^2, P(4,x) = 1 + 7*x + 10*x^2 + 2*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 4*x + 2*x^2;
P[4, x_] := 1 + 7*x + 10*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*x + 2*x^2)*P[n - 3, x] - (1 + x^2)*P[n - 2, x] + (2 + 2*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] = 1 + x; v[3] = 1 + 4*x + 2*x^2; v[4] = 1 + 7*x + 10*x^2 + 2*x^3; for(n = 5, N, my(i = n); v[i] = -x^2*v[i - 4] - (2*x + 2*x^2)*v[i - 3] - (1 + x^2)*v[i - 2] + (2 + 2*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0,…?
  - Laguerre’s theorem on differential operators [SP14] 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
  - Laguerre’s theorem on differential operators [SP14] 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)

Related ?

Same fingerprint ?

A122542 → D+1

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.