A144439 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-4\,x^{2} - 8\,n x^{2}\right)\,P_{n-4}(x) + \left(-4\,x + 12\,x^{2} - 8\,n x + 32\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-4\,x^{2} + 4\,x^{3} - 8\,n x^{2} + 8\,n x^{3}\right)\,P_{n-3}'(x) + \left(-3 + 6\,x + 5\,x^{2} - 6\,n + 24\,n x - 22\,n x^{2} - 8\,n^{2} x + 28\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{3} - 4\,n x + 12\,n x^{2} - 8\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 6\,x + 8\,n - 16\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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

\(P_{3}(x) = 31 + 62\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	8	2						10
3	31	62	3					96
4	102	652	306	4				1064
5	317	4812	7218	1268	5			13620
6	964	30174	103008	60348	4820	6		199320
7	2907	173346	1142841	1523788	433365	17442	7	3293696

All 15 rows

Technical details

Node id: n-32fce1bd1e1cbfb9 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ec24c0ec2e60f511

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:     8,       2;
  3:    31,      62,        3;
  4:   102,     652,      306,         4;
  5:   317,    4812,     7218,      1268,         5;
  6:   964,   30174,   103008,     60348,      4820,         6;
  7:  2907,  173346,  1142841,   1523788,    433365,     17442,        7;
  8:  8738,  946168, 10950618,  27702872,  18251030,   2838504,    61166,      8;
  9: 26233, 5004608, 95624640, 412982488, 516228110, 191249280, 17516128, 209864, 9;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = -(8*n+4)*x^2*P(n-4,x) - ((8*n+4)*x+(16*n^2-32*n-12)*x^2)*P(n-3,x) - ((8*n+4)*x^2-(8*n+4)*x^3)*d/dx P(n-3,x) - (6*n+3+(8*n^2-24*n-6)*x+(8*n^3-28*n^2+22*n-5)*x^2)*P(n-2,x) - ((4*n+2)*x+(8*n^2-12*n)*x^2-(8*n^2-8*n+2)*x^3)*d/dx P(n-2,x) + (8*n-2+(8*n^2-16*n+6)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 31 + 62*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 2*x;
P[3, x_] := 31 + 62*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 4)*x^2*P[n - 4, x] - ((8*n + 4)*x + (16*n^2 - 32*n - 12)*x^2)*P[n - 3, x] - ((8*n + 4)*x^2 - (8*n + 4)*x^3)*D[P[n - 3, x], x] - (6*n + 3 + (8*n^2 - 24*n - 6)*x + (8*n^3 - 28*n^2 + 22*n - 5)*x^2)*P[n - 2, x] - ((4*n + 2)*x + (8*n^2 - 12*n)*x^2 - (8*n^2 - 8*n + 2)*x^3)*D[P[n - 2, x], x] + (8*n - 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = 8 + 2*x; v[4] = 31 + 62*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n + 4)*x^2*v[i - 4] - ((8*n + 4)*x + (16*n^2 - 32*n - 12)*x^2)*v[i - 3] - ((8*n + 4)*x^2 - (8*n + 4)*x^3)*deriv(v[i - 3], x) - (6*n + 3 + (8*n^2 - 24*n - 6)*x + (8*n^3 - 28*n^2 + 22*n - 5)*x^2)*v[i - 2] - ((4*n + 2)*x + (8*n^2 - 12*n)*x^2 - (8*n^2 - 8*n + 2)*x^3)*deriv(v[i - 2], x) + (8*n - 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.