A185410 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(3 - 12\,x + 24\,x^{2} - 3\,n + 16\,n x - 44\,n x^{2} - 4\,n^{2} x + 24\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 10\,x^{2} + 8\,x^{3} - 2\,n x + 14\,n x^{2} - 12\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-7 + 16\,x + 4\,n - 16\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	P_n(1)
0							0
1							0
2	2						2
3	10	8					18
4	36	120	24				180
5	116	1032	888	64			2100
6	358	7016	15504	5312	160		28350
7	1086	42240	192720	171840	28320	384	436590

All 15 rows

Technical details

Node id: n-a1002d4e4f315530 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8ee1ad9f8b480ebb

Observed on the stored rows ?

Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
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:    0;
  1:    0;
  2:    2;
  3:   10,       8;
  4:   36,     120,       24;
  5:  116,    1032,      888,       64;
  6:  358,    7016,    15504,     5312,      160;
  7: 1086,   42240,   192720,   171840,    28320,      384;
  8: 3272,  237264,  1982640,  3602240,  1600320,   140928,    896;
  9: 9832, 1277936, 18149232, 58855360, 53625920, 13354368, 671104, 2048;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(3*n-3+(4*n^2-16*n+12)*x+(4*n^3-24*n^2+44*n-24)*x^2)*P(n-2,x) - ((2*n-2)*x+(4*n^2-14*n+10)*x^2-(4*n^2-12*n+8)*x^3)*d/dx P(n-2,x) + (4*n-7+(4*n^2-16*n+16)*x)*P(n-1,x) + ((2*n-4)*x-(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (4*n^2 - 16*n + 12)*x + (4*n^3 - 24*n^2 + 44*n - 24)*x^2)*P[n - 2, x] - ((2*n - 2)*x + (4*n^2 - 14*n + 10)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], x] + (4*n - 7 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] + ((2*n - 4)*x - (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (4*n^2 - 16*n + 12)*x + (4*n^3 - 24*n^2 + 44*n - 24)*x^2)*v[i - 2] - ((2*n - 2)*x + (4*n^2 - 14*n + 10)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(v[i - 2], x) + (4*n - 7 + (4*n^2 - 16*n + 16)*x)*v[i - 1] + ((2*n - 4)*x - (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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 A decomposition of the double factorials A001147.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A triangular decomposition of the double factorial numbers A001147.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)

References

[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.