A375853 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(2 - 2\,x + 5\,n - 9\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 8\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	8							8
3	56	40						96
4	216	432	120					768
5	616	2376	1848	280				5120
6	1456	9152	13728	5824	560			30720
7	3024	28080	68640	56160	15120	1008		172032
8	5712	73440	265200	353600	183600	34272	1680	917504

All 15 rows

Technical details

Node id: n-a5a4d355964274d4 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-48ce0a28efd19480

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     0;
   2:     8;
   3:    56,     40;
   4:   216,    432,     120;
   5:   616,   2376,    1848,     280;
   6:  1456,   9152,   13728,    5824,     560;
   7:  3024,  28080,   68640,   56160,   15120,    1008;
   8:  5712,  73440,  265200,  353600,  183600,   34272,    1680;
   9: 10032, 170544,  852720, 1679600, 1421200,  511632,   70224,   2640;
  10: 16632, 361152, 2387616, 6511680, 8139600, 4775232, 1264032, 133056, 3960;

%F Formula

The row polynomials satisfy (2*n^2-5*n+2)*P(n,x) = (2*n^2+5*n+2+(6*n^2-9*n-2)*x)*P(n-1,x) + (4*n*x-4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 8.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 5*n + 2 + (6*n^2 - 9*n - 2)*x)*P[n - 1, x] + (4*n*x - 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 8; for(n = 3, N, my(i = n); v[i] = ((2*n^2 + 5*n + 2 + (6*n^2 - 9*n - 2)*x)*v[i - 1] + (4*n*x - 4*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
    
    ↑ 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.