A208509 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(16 - x - 9\,n + 5\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(24 + 40\,x - 2\,x^{2} - 12\,n - 14\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(12\,x + 12\,x^{2}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	6	1							7
3	12	5							17
4	24	16	1						41
5	48	44	7						99
6	96	112	30	1					239
7	192	272	104	9					577
8	384	640	320	48	1				1393
9	768	1472	912	200	11				3363
10	1536	3328	2464	720	70	1			8119
11	3072	7424	6400	2352	340	13			19601
12	6144	16384	16128	7168	1400	96	1		47321
13	12288	35840	39680	20736	5152	532	15		114243
14	24576	77824	95744	57600	17472	2464	126	1	275807

Technical details

Node id: n-c6da825af22846f3 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-3e306556aac30f2d

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:   3;
  2:   6,    1;
  3:  12,    5;
  4:  24,   16,   1;
  5:  48,   44,   7;
  6:  96,  112,  30,   1;
  7: 192,  272, 104,   9;
  8: 384,  640, 320,  48,  1;
  9: 768, 1472, 912, 200, 11;

%F Formula

The row polynomials satisfy (n^2+2*n+1)*P(n,x) = (5*n^2-9*n+16+(n^2-1)*x)*P(n-1,x) - (12*n-24+(14*n-40)*x+(2*n+2)*x^2)*d/dx P(n-1,x) + (12*x+12*x^2)*d^2/dx^2 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[((5*n^2 - 9*n + 16 + (n^2 - 1)*x)*P[n - 1, x] - (12*n - 24 + (14*n - 40)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x] + (12*x + 12*x^2)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n + 1)]];
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] = ((5*n^2 - 9*n + 16 + (n^2 - 1)*x)*v[i - 1] - (12*n - 24 + (14*n - 40)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x) + (12*x + 12*x^2)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n + 1)); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Candidate equalities ?

A208509 → basis A007318 15 rows; no all-index equality certificate was found

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