A105070 → D+1

nonnegative coefficients

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

Recurrence

\(\left(5\,n - 12\,n x + 4\,n x^{2} - 3\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(3 - 14\,x + 20\,x^{2} - 8\,x^{3} + 2\,n - 20\,n x^{2} + 8\,n x^{3} - 4\,n^{2} x + 8\,n^{2} x^{2} + n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 12\,x^{2} + 24\,x^{3} - 16\,x^{4} - 2\,n x + 12\,n x^{2} - 16\,n x^{3} + 2\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	5	2							7
3	12	8							20
4	25	28	4						57
5	46	88	24						158
6	77	238	108	8					431
7	120	560	416	64					1160
8	177	1176	1368	352	16				3089
9	250	2256	3888	1600	160				8154
10	341	4026	9768	6160	1040	32			21367
11	452	6776	22176	20416	5440	384			55644
12	585	10868	46332	59488	23920	2880	64		144137
13	742	16744	90376	155584	90272	17024	896		371638
14	925	24934	166452	371800	298480	84000	7616	128	954335

Technical details

Node id: n-c28e607bfa1726cb · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-20fb9c15ea50c91a

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   5,    2;
  3:  12,    8;
  4:  25,   28,    4;
  5:  46,   88,   24;
  6:  77,  238,  108,    8;
  7: 120,  560,  416,   64;
  8: 177, 1176, 1368,  352,  16;
  9: 250, 2256, 3888, 1600, 160;

%F Formula

The row polynomials satisfy (n^3-3*n^2+5*n+(4*n^2-12*n)*x+4*n*x^2)*P(n,x) = (n^3+2*n+3+(2*n^3-4*n^2-14)*x+(8*n^2-20*n+20)*x^2+(8*n-8)*x^3)*P(n-1,x) + ((2*n^2-2*n+2)*x-(4*n^2-12*n+12)*x^2-(16*n-24)*x^3-16*x^4)*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[((n^3 + 2*n + 3 + (2*n^3 - 4*n^2 - 14)*x + (8*n^2 - 20*n + 20)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 2)*x - (4*n^2 - 12*n + 12)*x^2 - (16*n - 24)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 5*n + (4*n^2 - 12*n)*x + 4*n*x^2)]];
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] = ((n^3 + 2*n + 3 + (2*n^3 - 4*n^2 - 14)*x + (8*n^2 - 20*n + 20)*x^2 + (8*n - 8)*x^3)*v[i - 1] + ((2*n^2 - 2*n + 2)*x - (4*n^2 - 12*n + 12)*x^2 - (16*n - 24)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 5*n + (4*n^2 - 12*n)*x + 4*n*x^2)); 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 T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.?
  - 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.