A143685 → γ → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n + 9\,n^{2} x\right)\,P_{n}(x) = \left(-36\,x - 324\,x^{2} + n + 36\,n x + 162\,n x^{2} + 9\,n^{2} x + 162\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-x - 54\,x^{2} - 648\,x^{3} - 9\,n x^{2} - 324\,n x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	18							19
3	1	36							37
4	1	54	243						298
5	1	72	729						802
6	1	90	1458	2916					4465
7	1	108	2430	11664					14203
8	1	126	3645	29160	32805				65737
9	1	144	5103	58320	164025				227593
10	1	162	6804	102060	492075	354294			955396
11	1	180	8748	163296	1148175	2125764			3446164
12	1	198	10935	244944	2296350	7440174	3720087		13712689
13	1	216	13365	349920	4133430	19840464	26040609		50378005
14	1	234	16038	481140	6889050	44641044	104162436	38263752	194453695

Technical details

Node id: n-b4e5e37f3793a6df · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e0fc085c281e048d

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: 1,  18;
  3: 1,  36;
  4: 1,  54,  243;
  5: 1,  72,  729;
  6: 1,  90, 1458,  2916;
  7: 1, 108, 2430, 11664;
  8: 1, 126, 3645, 29160,  32805;
  9: 1, 144, 5103, 58320, 164025;

%F Formula

The row polynomials satisfy (n+9*n^2*x)*P(n,x) = (n+(9*n^2+36*n-36)*x+(162*n^2+162*n-324)*x^2)*P(n-1,x) - (x+(9*n+54)*x^2+(324*n+648)*x^3)*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 + (9*n^2 + 36*n - 36)*x + (162*n^2 + 162*n - 324)*x^2)*P[n - 1, x] - (x + (9*n + 54)*x^2 + (324*n + 648)*x^3)*D[P[n - 1, x], x])/(n + 9*n^2*x)]];
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 + (9*n^2 + 36*n - 36)*x + (162*n^2 + 162*n - 324)*x^2)*v[i - 1] - (x + (9*n + 54)*x^2 + (324*n + 648)*x^3)*deriv(v[i - 1], x))/(n + 9*n^2*x)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.