A035342 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(2\,x + 2\,n + 2\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(3\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	15	18	3						36
3	105	174	54	4					337
4	945	1950	855	120	5				3875
5	10395	25290	14040	2820	225	6			52776
6	135135	374850	248535	63840	7350	378	7		830095
7	2027025	6267870	4787370	1480500	219450	16380	588	8	14799191

All 15 rows

Technical details

Node id: n-2742cd6041d47f22 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-93df90a20065ab4f

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 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:         3,          2;
  2:        15,         18,          3;
  3:       105,        174,         54,         4;
  4:       945,       1950,        855,       120,         5;
  5:     10395,      25290,      14040,      2820,       225,        6;
  6:    135135,     374850,     248535,     63840,      7350,      378,       7;
  7:   2027025,    6267870,    4787370,   1480500,    219450,    16380,     588,     8;
  8:  34459425,  116875710,  100361835,  35993160,   6459075,   623700,   32634,   864,    9;
  9: 654729075, 2406433050, 2282912100, 925665300, 193550175, 22719690, 1543500, 59760, 1215, 10;

%F Formula

The row polynomials satisfy 2*n*P(n,x) = (4*n^2+2*n+(2*n+2)*x)*P(n-1,x) + (2*n+3)*x*d/dx P(n-1,x) + x*d^2/dx^2 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[((4*n^2 + 2*n + (2*n + 2)*x)*P[n - 1, x] + (2*n + 3)*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2*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] = ((4*n^2 + 2*n + (2*n + 2)*x)*v[i - 1] + (2*n + 3)*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2*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 (−∞, 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 The convolution matrix of the double factorial of odd numbers (A001147).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
    
    Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.