A107131 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	3	1						6
3	4	9	6	1					20
4	9	26	26	10	1				72
5	21	75	100	60	15	1			272
6	51	216	360	295	120	21	1		1064
7	127	623	1246	1295	735	217	28	1	4272

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

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

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

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

Technical details

Node id: n-9a0d716b5162dd54 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-e0858ae2787f914b)
Fingerprint: f6-5395a3bd8e193ab8

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:   1,    1;
  2:   2,    3,     1;
  3:   4,    9,     6,     1;
  4:   9,   26,    26,    10,     1;
  5:  21,   75,   100,    60,    15,     1;
  6:  51,  216,   360,   295,   120,    21,    1;
  7: 127,  623,  1246,  1295,   735,   217,   28,   1;
  8: 323, 1800,  4200,  5292,  3864,  1624,  364,  36,  1;
  9: 835, 5211, 13896, 20580, 18396, 10080, 3276, 576, 45, 1;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(n-4-3*n*x)*P(n-1,x) + (6+4*x-2*x^2)*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 - 4 - 3*n*x)*P[n - 1, x] + (6 + 4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 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 - 4 - 3*n*x)*v[i - 1] + (6 + 4*x - 2*x^2)*deriv(v[i - 1], x))/(n + 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 (−∞, -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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from A Motzkin related triangle.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from A Motzkin related triangle.?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

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