A257613 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-24 + 16\,x - 24\,n + 4\,n x^{2} - 16\,n^{2} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x + 8\,x^{2} - 8\,n x + 4\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 10\,n + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	48	32													80
3	416	832	192												1440
4	3136	13312	9408	1024											26880
5	21888	169216	253824	87552	5120										537600
6	145664	1879040	5076480	3758080	728320	24576									11612160
7	939520	19110912	84587520	112783360	47777280	5637120	114688								270950400
8	5932032	182951936	1247204352	2706800640	2078673920	548855808	41524224	524288							6812467200
9	36902912	1677168640	16862889984	55762755584	69703444480	33725779968	5870090240	295223296	2359296						183936614400
10	227184640	14893465600	213912453120	1028894720000	1951696445440	1543342080000	499129057280	59573862400	2044661760	10485760					5313724416000
11	1388273664	129143848960	2585928499200	17480635514880	47901644636160	57481973563392	30591112151040	6895809331200	581147320320	13882736640	46137344				163662712012800
12	8438693888	1099787927552	30121032007680	278725719490560	1063937323991040	1839423154028544	1489512253587456	557451438981120	90363096023040	5498939637760	92825632768	201326592			5356234211328000
13	51101925376	9237115502592	340802685468672	4228258906112000	21863265523138560	52411816662663168	61147119439773696	34981224837021696	9513582538752000	1136008951562240	50804135264256	613223104512	872415232		185682785992704000
14	308624818176	76758631841792	3768274359287808	61644792808341504	422362837242019840	1363307439157936128	2201296299831853056	1817743252210581504	760253107035635712	154111982020853760	13817005984055296	460551791050752	4012122636288	3758096384	6798846625579008000

Technical details

Node id: n-1d5d686e65769839 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-96547810142a253d

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 -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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:        4;
  2:       48,         32;
  3:      416,        832,         192;
  4:     3136,      13312,        9408,        1024;
  5:    21888,     169216,      253824,       87552,        5120;
  6:   145664,    1879040,     5076480,     3758080,      728320,       24576;
  7:   939520,   19110912,    84587520,   112783360,    47777280,     5637120,     114688;
  8:  5932032,  182951936,  1247204352,  2706800640,  2078673920,   548855808,   41524224,    524288;
  9: 36902912, 1677168640, 16862889984, 55762755584, 69703444480, 33725779968, 5870090240, 295223296, 2359296;

%F Formula

The row polynomials satisfy n*P(n,x) = -(24*n+24+(16*n^2-16)*x+(4*n^3-4*n)*x^2)*P(n-2,x) - ((8*n+8)*x+(4*n^2-4*n-8)*x^2-(4*n^2+4*n)*x^3)*d/dx P(n-2,x) + (10*n+4+4*n^2*x)*P(n-1,x) + (2*n*x-2*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n + 24 + (16*n^2 - 16)*x + (4*n^3 - 4*n)*x^2)*P[n - 2, x] - ((8*n + 8)*x + (4*n^2 - 4*n - 8)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (10*n + 4 + 4*n^2*x)*P[n - 1, x] + (2*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n + 24 + (16*n^2 - 16)*x + (4*n^3 - 4*n)*x^2)*v[i - 2] - ((8*n + 8)*x + (4*n^2 - 4*n - 8)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (10*n + 4 + 4*n^2*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.