A257611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(15 - 84\,x + 107\,x^{2} - 46\,x^{3} + 8\,x^{4} - 30\,n + 60\,n x - 82\,n x^{2} + 56\,n x^{3} - 16\,n x^{4} - 24\,n^{2} x + 28\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 36\,x^{2} + 58\,x^{3} - 44\,x^{4} + 16\,x^{5} - 12\,n x + 28\,n x^{2} - 40\,n x^{3} + 24\,n x^{4} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-18 + 44\,x - 26\,x^{2} + 8\,x^{3} + 16\,n - 20\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 14\,x^{2} - 8\,x^{3} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	3							9
2	39	48	9						96
3	240	639	294	27					1200
4	1389	7272	6906	1632	81				17280
5	7674	72723	130584	62370	8646	243			282240
6	41043	660840	2073147	1828680	511833	44688	729		5160960
7	214692	5604351	28800558	43507725	22181160	3970953	227814	2187	104509440

All 15 rows

Technical details

Node id: n-aff6b1b26c2ca304 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9059ca5320a7cf25

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:       1;
  1:       6,         3;
  2:      39,        48,          9;
  3:     240,       639,        294,          27;
  4:    1389,      7272,       6906,        1632,          81;
  5:    7674,     72723,     130584,       62370,        8646,         243;
  6:   41043,    660840,    2073147,     1828680,      511833,       44688,        729;
  7:  214692,   5604351,   28800558,    43507725,    22181160,     3970953,     227814,      2187;
  8: 1106265,  45217776,  362141748,   879096864,   758238510,   245715120,   29756964,   1152192,    6561;
  9: 5642862, 351839619, 4226509248, 15668996244, 21476793852, 11686267698, 2562891216, 218093652, 5800326, 19683;

%F Formula

The row polynomials satisfy (2*n-3+4*x)*P(n,x) = -(30*n-15+(24*n^2-60*n+84)*x+(8*n^3-28*n^2+82*n-107)*x^2+(24*n^2-56*n+46)*x^3+(16*n-8)*x^4)*P(n-2,x) - ((12*n-6)*x+(8*n^2-28*n+36)*x^2-(8*n^2-40*n+58)*x^3-(24*n-44)*x^4-16*x^5)*d/dx P(n-2,x) + (16*n-18+(8*n^2-20*n+44)*x+(20*n-26)*x^2+8*x^3)*P(n-1,x) + ((4*n-6)*x-(4*n-14)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 15 + (24*n^2 - 60*n + 84)*x + (8*n^3 - 28*n^2 + 82*n - 107)*x^2 + (24*n^2 - 56*n + 46)*x^3 + (16*n - 8)*x^4)*P[n - 2, x] - ((12*n - 6)*x + (8*n^2 - 28*n + 36)*x^2 - (8*n^2 - 40*n + 58)*x^3 - (24*n - 44)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (16*n - 18 + (8*n^2 - 20*n + 44)*x + (20*n - 26)*x^2 + 8*x^3)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 14)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n - 3 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n - 15 + (24*n^2 - 60*n + 84)*x + (8*n^3 - 28*n^2 + 82*n - 107)*x^2 + (24*n^2 - 56*n + 46)*x^3 + (16*n - 8)*x^4)*v[i - 2] - ((12*n - 6)*x + (8*n^2 - 28*n + 36)*x^2 - (8*n^2 - 40*n + 58)*x^3 - (24*n - 44)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (16*n - 18 + (8*n^2 - 20*n + 44)*x + (20*n - 26)*x^2 + 8*x^3)*v[i - 1] + ((4*n - 6)*x - (4*n - 14)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n - 3 + 4*x)); 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, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.?
  - 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) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.?
    
    ↑ 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.