A115951 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n x - 6\,n x^{2} + 4\,n x^{3} - n^{2} + 2\,n^{2} x + 4\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12\,x - 16\,x^{2} + 4\,x^{3} + 8\,x^{4} - 6\,n + 2\,n x + 16\,n x^{2} + 8\,n^{2} + 6\,n^{2} x - 8\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 24\,x^{4} + 16\,x^{5} - 4\,n x + 4\,n x^{2} + 24\,n x^{3} + 16\,n x^{4} + 4\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	2	14	6													22
3	0	24	72	20												116
4	0	12	186	340	70											608
5	0	0	180	1180	1540	252										3152
6	0	0	60	1700	6720	6804	924									16208
7	0	0	0	1120	12880	35784	29568	3432								82784
8	0	0	0	280	12670	85680	182028	126984	12870							420512
9	0	0	0	0	6300	112140	522984	895752	540540	48620						2126336
10	0	0	0	0	1260	83412	854700	3003000	4298580	2285140	184756					10710848
11	0	0	0	0	0	33264	846384	5885880	16473600	20225920	9607312	705432				53777792
12	0	0	0	0	0	5544	505428	7279272	37657620	87224280	93671292	40209624	2704156			269257216
13	0	0	0	0	0	0	168168	5789784	55855800	227833320	448957080	428197224	167657672	10400600		1344859648
14	0	0	0	0	0	0	24024	2886312	55495440	394113720	1319157840	2258087832	1936175696	696840200	40116600	6702897664

Technical details

Node id: n-27decca1d071fa2c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8175414b3da05124

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 0)
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: 1;
  1: 2,  2;
  2: 2, 14,   6;
  3: 0, 24,  72,   20;
  4: 0, 12, 186,  340,    70;
  5: 0,  0, 180, 1180,  1540,    252;
  6: 0,  0,  60, 1700,  6720,   6804,    924;
  7: 0,  0,   0, 1120, 12880,  35784,  29568,   3432;
  8: 0,  0,   0,  280, 12670,  85680, 182028, 126984,  12870;
  9: 0,  0,   0,    0,  6300, 112140, 522984, 895752, 540540, 48620;

%F Formula

The row polynomials satisfy (n^3-n^2+(2*n^2-4*n)*x+(4*n^2-6*n)*x^2+4*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(6*n^2+2*n-12)*x+(8*n^2-16*n+16)*x^2-4*x^3-8*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(4*n^2+4*n-8)*x^2+24*n*x^3+(16*n+24)*x^4+16*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 8*n^2 + 6*n - (6*n^2 + 2*n - 12)*x + (8*n^2 - 16*n + 16)*x^2 - 4*x^3 - 8*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (4*n^2 + 4*n - 8)*x^2 + 24*n*x^3 + (16*n + 24)*x^4 + 16*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 + (2*n^2 - 4*n)*x + (4*n^2 - 6*n)*x^2 + 4*n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 - 8*n^2 + 6*n - (6*n^2 + 2*n - 12)*x + (8*n^2 - 16*n + 16)*x^2 - 4*x^3 - 8*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (4*n^2 + 4*n - 8)*x^2 + 24*n*x^3 + (16*n + 24)*x^4 + 16*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 + (2*n^2 - 4*n)*x + (4*n^2 - 6*n)*x^2 + 4*n*x^3)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.