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	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	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
9	10032	170544	852720	1679600	1421200	511632	70224	2640							4718592
10	16632	361152	2387616	6511680	8139600	4775232	1264032	133056	3960						23592960
11	26312	710424	5999136	21633248	37442160	32449872	13997984	2841696	236808	5720					115343360
12	40040	1315600	13813800	63627200	145608400	174730080	111347600	36836800	5920200	400400	8008				553648128
13	58968	2316600	29601000	169533000	495558000	786285360	693781200	339066000	88803000	11583000	648648	10920			2617245696
14	84448	3908736	59716800	416208000	1512756000	3065852160	3576827520	2420409600	936468000	199056000	21498048	1013376	14560		12213813248
15	118048	6357728	114439104	953659200	4218108000	10629632160	15840236160	14172842880	7592594400	2384148000	419610048	38146368	1534624	19040	56371445760

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.