A161198 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 24\,x + 36\,x^{2} + 12\,n - 16\,n x - 4\,n^{2}\right)\,P_{n-2}(x) + \left(-24\,x^{2} + 16\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-4 + 8\,x + 4\,n\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	8							11
3	15	64	36						115
4	105	568	540	128					1341
5	945	5952	7416	3072	400				17785
6	10395	73032	108360	60160	14000	1152			267099
7	135135	1032576	1737324	1167360	370000	55296	3136		4500827
8	2027025	16569720	30706452	23687552	9198000	1895040	197568	8192	84289549

All 15 rows

Technical details

Node id: n-ff64fee400d5c844 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-bc4fc28a66e5f545

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 -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:         3,          8;
   3:        15,         64,          36;
   4:       105,        568,         540,         128;
   5:       945,       5952,        7416,        3072,        400;
   6:     10395,      73032,      108360,       60160,      14000,       1152;
   7:    135135,    1032576,     1737324,     1167360,     370000,      55296,      3136;
   8:   2027025,   16569720,    30706452,    23687552,    9198000,    1895040,    197568,     8192;
   9:  34459425,  297901440,   596573424,   511866880,  230389600,   58705920,   8517376,   655360,   20736;
  10: 654729075, 5935802760, 12675451536, 11846620672, 5976986400, 1778934528, 321640704, 34701312, 2052864, 51200;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(16*n-24)*x-36*x^2)*P(n-2,x) + ((16*n-24)*x^2+16*x^3)*d/dx P(n-2,x) + (4*n-4+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 8*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (16*n - 24)*x - 36*x^2)*P[n - 2, x] + ((16*n - 24)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (4*n - 4 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 8*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (16*n - 24)*x - 36*x^2)*v[i - 2] + ((16*n - 24)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (4*n - 4 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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 basis_map_A007318 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.