A128813 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 4\,n x - n^{2} - 2\,n^{2} x - n^{2} x^{2} - 3\,n^{3} - 2\,n^{3} x + n^{3} x^{2} + n^{4} + 2\,n^{4} x + n^{4} x^{2} - n^{5} - 2\,n^{5} x - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(4\,n - 4\,n x + 4\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	1							4
2	18	16	3						37
3	174	280	135	18					607
4	2474	6000	5095	1728	180				15477
5	48824	160980	202585	119193	31500	2700			565782
6	1281104	5385192	9118576	7867584	3574053	777600	56700		28060809
7	43223312	221841096	475716960	548422480	362777373	134871084	25004700	1587600	1813444605

All 15 rows

Technical details

Node id: n-387e63ced86a0d84 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-229aa602b5cf76cb

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,            1;
  2:          18,           16,             3;
  3:         174,          280,           135,            18;
  4:        2474,         6000,          5095,          1728,           180;
  5:       48824,       160980,        202585,        119193,         31500,          2700;
  6:     1281104,      5385192,       9118576,       7867584,       3574053,        777600,         56700;
  7:    43223312,    221841096,     475716960,     548422480,     362777373,     134871084,      25004700,      1587600;
  8:  1825635152,  11093826408,   28916113392,   42016610368,   36951982389,   19905010560,    6280626924,   1016064000,    57153600;
  9: 94448950112, 664153445328, 2043132574212, 3596474490508, 3972604195359, 2834624426985, 1291229437284, 354514739580, 50923857600, 2571912000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(2*n^5-2*n^4+2*n^3+2*n^2-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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 coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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 coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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.