A122542 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{3}(x) = 11 + 16\,x + 4\,x^{2}\)

\(P_{4}(x) = 33 + 82\,x + 52\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	2														5
3	11	16	4													31
4	33	82	52	8												175
5	97	336	368	144	16											961
6	275	1226	1936	1312	368	32										5149
7	763	4160	8612	8496	4112	896	64									27103
8	2081	13418	34420	44936	31728	11872	2112	128								140695
9	5601	41696	127680	208096	195232	106496	32384	4864	256							722305
10	14915	125906	448256	877312	1031776	748864	331392	84736	11008	512						3674677
11	39371	371632	1508452	3448864	4888736	4457216	2624512	974592	214784	24576	1024					18553759
12	103169	1076802	4908244	12843592	21335904	23497664	17395712	8589824	2743552	530944	54272	2048				93081727
13	268673	3072432	15539184	45808112	87332016	112954368	101135232	62799104	26646272	7458816	1286144	118784	4096			464423233
14	695955	8653594	48090192	157717920	339538384	504962400	531684224	399528192	213126400	79169024	19709952	3063808	258048	8192		2306206285
15	1794427	24104288	145999940	527312848	1265521328	2128956032	2580920128	2282502400	1473759488	687935488	227027968	50868224	7196672	557056	16384	11404472671

Technical details

Node id: n-9dd9e585db514cc1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e553b6ffd2273fac

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:     3,      2;
   3:    11,     16,      4;
   4:    33,     82,     52,      8;
   5:    97,    336,    368,    144,      16;
   6:   275,   1226,   1936,   1312,     368,     32;
   7:   763,   4160,   8612,   8496,    4112,    896,     64;
   8:  2081,  13418,  34420,  44936,   31728,  11872,   2112,   128;
   9:  5601,  41696, 127680, 208096,  195232, 106496,  32384,  4864,   256;
  10: 14915, 125906, 448256, 877312, 1031776, 748864, 331392, 84736, 11008, 512;

%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) = 1, P(2,x) = 3 + 2*x, P(3,x) = 11 + 16*x + 4*x^2, P(4,x) = 33 + 82*x + 52*x^2 + 8*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 11 + 16*x + 4*x^2;
P[4, x_] := 33 + 82*x + 52*x^2 + 8*x^3;
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] = 1; v[2] = 3 + 2*x; v[3] = 11 + 16*x + 4*x^2; v[4] = 33 + 82*x + 52*x^2 + 8*x^3; 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.