A086645 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-18 + 24\,x + 9\,n - 104\,n x + 90\,n^{2} x - 8\,n^{2} x^{2} - 22\,n^{3} x + 20\,n^{3} x^{2} - 16\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-18 - 264\,x - 192\,x^{2} + 9\,n + 40\,n x + 16\,n x^{2} + 90\,n^{2} x + 336\,n^{2} x^{2} - 32\,n^{2} x^{3} - 22\,n^{3} x - 132\,n^{3} x^{2} - 48\,n^{3} x^{3} - 16\,n^{4} x^{2} + 4\,n^{5} x^{2} + 16\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(18\,x - 48\,x^{2} - 96\,x^{3} + 4\,n x^{2} - 16\,n x^{3} - 44\,n^{2} x^{2} + 184\,n^{2} x^{3} - 32\,n^{2} x^{4} + 16\,n^{3} x^{3} - 64\,n^{3} x^{4} + 8\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1								1
3	1	16							17
4	1	48							49
5	1	96	144						241
6	1	160	720						881
7	1	240	2160	1024					3425
8	1	336	5040	7168					12545
9	1	448	10080	28672	6400				45601
10	1	576	18144	86016	57600				162337
11	1	720	30240	215040	288000	36864			570865
12	1	880	47520	473088	1056000	405504			1982993
13	1	1056	71280	946176	3168000	2433024	200704		6820241
14	1	1248	102960	1757184	8236800	10543104	2609152		23250449
15	1	1456	144144	3075072	19219200	36900864	18264064	1048576	78653377

Technical details

Node id: n-dcd5a5e04c1e6b0a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-104b93e0fc2039fc

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1;
   3: 1,  16;
   4: 1,  48;
   5: 1,  96,   144;
   6: 1, 160,   720;
   7: 1, 240,  2160,  1024;
   8: 1, 336,  5040,  7168;
   9: 1, 448, 10080, 28672,  6400;
  10: 1, 576, 18144, 86016, 57600;

%F Formula

The row polynomials satisfy (9*n-18-(22*n^3-90*n^2+104*n-24)*x+(4*n^5-16*n^4+20*n^3-8*n^2)*x^2)*P(n,x) = (9*n-18-(22*n^3-90*n^2-40*n+264)*x+(4*n^5-16*n^4-132*n^3+336*n^2+16*n-192)*x^2+(16*n^5-48*n^3-32*n^2)*x^3)*P(n-1,x) + (18*x-(44*n^2-4*n+48)*x^2+(8*n^4+16*n^3+184*n^2-16*n-96)*x^3-(32*n^4+64*n^3+32*n^2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 18 - (22*n^3 - 90*n^2 - 40*n + 264)*x + (4*n^5 - 16*n^4 - 132*n^3 + 336*n^2 + 16*n - 192)*x^2 + (16*n^5 - 48*n^3 - 32*n^2)*x^3)*P[n - 1, x] + (18*x - (44*n^2 - 4*n + 48)*x^2 + (8*n^4 + 16*n^3 + 184*n^2 - 16*n - 96)*x^3 - (32*n^4 + 64*n^3 + 32*n^2)*x^4)*D[P[n - 1, x], x])/(9*n - 18 - (22*n^3 - 90*n^2 + 104*n - 24)*x + (4*n^5 - 16*n^4 + 20*n^3 - 8*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((9*n - 18 - (22*n^3 - 90*n^2 - 40*n + 264)*x + (4*n^5 - 16*n^4 - 132*n^3 + 336*n^2 + 16*n - 192)*x^2 + (16*n^5 - 48*n^3 - 32*n^2)*x^3)*v[i - 1] + (18*x - (44*n^2 - 4*n + 48)*x^2 + (8*n^4 + 16*n^3 + 184*n^2 - 16*n - 96)*x^3 - (32*n^4 + 64*n^3 + 32*n^2)*x^4)*deriv(v[i - 1], x))/(9*n - 18 - (22*n^3 - 90*n^2 + 104*n - 24)*x + (4*n^5 - 16*n^4 + 20*n^3 - 8*n^2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows: T(n, k) = binomial(2n, 2k).?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A013609 → Veronese 2 → (θ+1)²

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023