A144441 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 4\,n\right)\,P_{n}(x) = \left(-16\,x^{2} - 64\,n x^{2}\right)\,P_{n-4}(x) + \left(-8\,x + 24\,x^{2} - 32\,n x + 320\,n x^{2} - 128\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-16\,x^{2} + 16\,x^{3} - 64\,n x^{2} + 64\,n x^{3}\right)\,P_{n-3}'(x) + \left(-5 - 2\,x + 99\,x^{2} - 20\,n + 112\,n x - 300\,n x^{2} - 32\,n^{2} x + 272\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x - 32\,x^{2} + 36\,x^{3} - 16\,n x + 112\,n x^{2} - 96\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-14 + 42\,x + 24\,n - 80\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 12\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 14 + 2\,x\)

\(P_{3}(x) = 83 + 166\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	14	2						16
3	83	166	3					252
4	432	3100	1296	4				4832
5	2181	39796	59694	8724	5			110400
6	10930	435774	1570644	871548	54650	6		2943552
7	54679	4398474	31769385	42359180	10996185	328074	7	89905984

All 15 rows

Technical details

Node id: n-bf0f00698b963e13 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b08b42757ceb90b6

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 -1)
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:       1;
  2:      14,         2;
  3:      83,       166,          3;
  4:     432,      3100,       1296,           4;
  5:    2181,     39796,      59694,        8724,           5;
  6:   10930,    435774,    1570644,      871548,       54650,           6;
  7:   54679,   4398474,   31769385,    42359180,    10996185,      328074,          7;
  8:  273428,  42407656,  554168580,  1448588888,   923614300,   127222968,    1913996,        8;
  9: 1367177, 397965160, 8820869484, 40312168056, 50390210070, 17641738968, 1392878060, 10937416, 9;

%F Formula

The row polynomials satisfy (4*n-3)*P(n,x) = -(64*n+16)*x^2*P(n-4,x) - ((32*n+8)*x+(128*n^2-320*n-24)*x^2)*P(n-3,x) - ((64*n+16)*x^2-(64*n+16)*x^3)*d/dx P(n-3,x) - (20*n+5+(32*n^2-112*n+2)*x+(64*n^3-272*n^2+300*n-99)*x^2)*P(n-2,x) - ((16*n+4)*x+(64*n^2-112*n+32)*x^2-(64*n^2-96*n+36)*x^3)*d/dx P(n-2,x) + (24*n-14+(32*n^2-80*n+42)*x)*P(n-1,x) + ((16*n-12)*x-(16*n-12)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 14 + 2*x, P(3,x) = 83 + 166*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 14 + 2*x;
P[3, x_] := 83 + 166*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(64*n + 16)*x^2*P[n - 4, x] - ((32*n + 8)*x + (128*n^2 - 320*n - 24)*x^2)*P[n - 3, x] - ((64*n + 16)*x^2 - (64*n + 16)*x^3)*D[P[n - 3, x], x] - (20*n + 5 + (32*n^2 - 112*n + 2)*x + (64*n^3 - 272*n^2 + 300*n - 99)*x^2)*P[n - 2, x] - ((16*n + 4)*x + (64*n^2 - 112*n + 32)*x^2 - (64*n^2 - 96*n + 36)*x^3)*D[P[n - 2, x], x] + (24*n - 14 + (32*n^2 - 80*n + 42)*x)*P[n - 1, x] + ((16*n - 12)*x - (16*n - 12)*x^2)*D[P[n - 1, x], x])/(4*n - 3)]];
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] = 1; v[3] = 14 + 2*x; v[4] = 83 + 166*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(64*n + 16)*x^2*v[i - 4] - ((32*n + 8)*x + (128*n^2 - 320*n - 24)*x^2)*v[i - 3] - ((64*n + 16)*x^2 - (64*n + 16)*x^3)*deriv(v[i - 3], x) - (20*n + 5 + (32*n^2 - 112*n + 2)*x + (64*n^3 - 272*n^2 + 300*n - 99)*x^2)*v[i - 2] - ((16*n + 4)*x + (64*n^2 - 112*n + 32)*x^2 - (64*n^2 - 96*n + 36)*x^3)*deriv(v[i - 2], x) + (24*n - 14 + (32*n^2 - 80*n + 42)*x)*v[i - 1] + ((16*n - 12)*x - (16*n - 12)*x^2)*deriv(v[i - 1], x))/(4*n - 3)); 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 Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.