A089627 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(6 + 24\,x - 3\,n - 12\,n x\right)\,P_{n-2}(x) + \left(3 - 2\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	1				1
3	3	6			9
4	7	18			25
5	19	72	54		145
6	51	240	270		561
7	141	810	1350	540	2841
8	393	2646	5670	3780	12489

All 15 rows

Technical details

Node id: n-a25c84ed241587e6 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-ea55a274263d0861

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:    3,     6;
   4:    7,    18;
   5:   19,    72,    54;
   6:   51,   240,   270;
   7:  141,   810,  1350,    540;
   8:  393,  2646,  5670,   3780;
   9: 1107,  8568, 22680,  22680,  5670;
  10: 3139, 27432, 86184, 113400, 51030;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(3*n-6+(12*n-24)*x)*P(n-2,x) - (2*n-3)*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[(-(3*n - 6 + (12*n - 24)*x)*P[n - 2, x] - (2*n - 3)*P[n - 1, x])/(-n + 1)]];
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] = (-(3*n - 6 + (12*n - 24)*x)*v[i - 2] - (2*n - 3)*v[i - 1])/(-n + 1)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.?
  - Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool 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
  - Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
    
    from gamma_polynomial of narayana?
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from narayana?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    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 narayana?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from narayana?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - 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 ?

Same fingerprint ?

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