A143685 → γ → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-18 - 18\,x + 19\,n + 18\,n x\right)\,P_{n-1}(x) + \left(-37 - 73\,x - 36\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	10	9							19
3	19	18							37
4	109	189	81						379
5	280	522	243						1045
6	1261	3204	2673	729					7867
7	3781	10422	9558	2916					26677
8	15130	50607	62451	33534	6561				168283
9	49159	178434	242271	145800	32805				648469
10	185329	770067	1259793	1009665	393660	59049			3677563
11	627760	2818404	5046138	4502304	2001105	354294			15350005
12	2295721	11416968	23314878	24927426	14631030	4428675	531441		81546139
13	7945561	42432444	94095756	110863404	73161711	25627266	3720087		357846229
14	28607050	165846645	406682370	545044140	429187815	197164611	48361131	4782969	1825676731

Other recurrences

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

\(P_{n}(x) = \left(9 + 9\,x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)

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

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

Technical details

Node id: n-9957432edab6d129 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-c81b83be35fe8f46)
Fingerprint: f6-4f014acb052a88e2

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:     1;
  2:    10,      9;
  3:    19,     18;
  4:   109,    189,     81;
  5:   280,    522,    243;
  6:  1261,   3204,   2673,    729;
  7:  3781,  10422,   9558,   2916;
  8: 15130,  50607,  62451,  33534,  6561;
  9: 49159, 178434, 242271, 145800, 32805;

%F Formula

The row polynomials satisfy n*P(n,x) = (19*n-18+(18*n-18)*x)*P(n-1,x) - (37+73*x+36*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[((19*n - 18 + (18*n - 18)*x)*P[n - 1, x] - (37 + 73*x + 36*x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((19*n - 18 + (18*n - 18)*x)*v[i - 1] - (37 + 73*x + 36*x^2)*deriv(v[i - 1], x))/(n)); 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from gamma_polynomial of Pascal-(1,9,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,9,1) array.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,9,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,9,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from gamma_polynomial of Pascal-(1,9,1) array.?
  - ↑ Gamma criterion (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