A049403 → D+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 3 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	3	5	1								9
1	3	15	9	1							28
2	0	30	45	14	1						90
3	0	30	150	105	20	1					306
4	0	0	315	525	210	27	1				1078
5	0	0	315	1785	1470	378	35	1			3984
6	0	0	0	3780	7245	3528	630	44	1		15228
7	0	0	0	3780	24570	23625	7560	990	54	1	60580

All 15 rows

Technical details

Node id: n-9191274e47be5847 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e826f8cd3bf9f5b7

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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: 3,  5,   1;
  1: 3, 15,   9,    1;
  2: 0, 30,  45,   14,     1;
  3: 0, 30, 150,  105,    20,      1;
  4: 0,  0, 315,  525,   210,     27,      1;
  5: 0,  0, 315, 1785,  1470,    378,     35,      1;
  6: 0,  0,   0, 3780,  7245,   3528,    630,     44,     1;
  7: 0,  0,   0, 3780, 24570,  23625,   7560,    990,    54,    1;
  8: 0,  0,   0,    0, 51975, 114345,  65835,  14850,  1485,   65,  1;
  9: 0,  0,   0,    0, 51975, 384615, 426195, 162855, 27225, 2145, 77, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 3 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 2*n + (n^2 - 5*n + 6)*x + 2*x^2 - 2*x^3)*v[i - 1] + ((2*n^2 + 2*n)*x + (4*n - 4)*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n^2 + n + (2*n - 4)*x + 2*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 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).?
  - 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.