A274498 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-8 + 28\,x - 8\,x^{2} + 14\,n - 22\,n x + 4\,n x^{2} - 7\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-240 + 336\,x - 96\,x^{2} + 336\,n - 342\,n x + 120\,n x^{2} - 24\,n x^{3} - 161\,n^{2} + 130\,n^{2} x - 52\,n^{2} x^{2} + 8\,n^{2} x^{3} + 31\,n^{3} - 24\,n^{3} x + 8\,n^{3} x^{2} - 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(-36 - 12\,x + 96\,x^{2} - 48\,x^{3} + 46\,n + 40\,n x - 84\,n x^{2} + 32\,n x^{3} - 16\,n x^{4} - 18\,n^{2} - 22\,n^{2} x + 20\,n^{2} x^{2} - 16\,n^{2} x^{3} + 2\,n^{3} + 2\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 6\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1								0
2	6							6
3	18							18
4	108	72						180
5	324	216						540
6	810	1944	648					3402
7	2430	5832	1944					10206
8	4536	19440	23328	5184				52488
9	13608	58320	69984	15552				157464
10	21870	136080	291600	233280	38880			721710
11	65610	408240	874800	699840	116640			2165130
12	96228	787320	2449440	3499200	2099520	279936		9211644
13	288684	2361960	7348320	10497600	6298560	839808		27634932
14	398034	4041576	16533720	34292160	36741600	17635968	1959552	111602610

Technical details

Node id: n-23e6139da45dbcd9 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4f4ca2dec4972ed6

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     0;
  1:     0;
  2:     6;
  3:    18;
  4:   108,    72;
  5:   324,   216;
  6:   810,  1944,   648;
  7:  2430,  5832,  1944;
  8:  4536, 19440, 23328,  5184;
  9: 13608, 58320, 69984, 15552;

%F Formula

The row polynomials satisfy (n^3-7*n^2+14*n-8+(4*n^2-22*n+28)*x+(4*n-8)*x^2)*P(n,x) = -(2*n^4-31*n^3+161*n^2-336*n+240-(2*n^4-24*n^3+130*n^2-342*n+336)*x-(8*n^3-52*n^2+120*n-96)*x^2-(8*n^2-24*n)*x^3)*P(n-1,x) + (2*n^3-18*n^2+46*n-36+(2*n^3-22*n^2+40*n-12)*x-(4*n^3-20*n^2+84*n-96)*x^2-(16*n^2-32*n+48)*x^3-16*n*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^4 - 31*n^3 + 161*n^2 - 336*n + 240 - (2*n^4 - 24*n^3 + 130*n^2 - 342*n + 336)*x - (8*n^3 - 52*n^2 + 120*n - 96)*x^2 - (8*n^2 - 24*n)*x^3)*P[n - 1, x] + (2*n^3 - 18*n^2 + 46*n - 36 + (2*n^3 - 22*n^2 + 40*n - 12)*x - (4*n^3 - 20*n^2 + 84*n - 96)*x^2 - (16*n^2 - 32*n + 48)*x^3 - 16*n*x^4)*D[P[n - 1, x], x])/(n^3 - 7*n^2 + 14*n - 8 + (4*n^2 - 22*n + 28)*x + (4*n - 8)*x^2)]];
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] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n^4 - 31*n^3 + 161*n^2 - 336*n + 240 - (2*n^4 - 24*n^3 + 130*n^2 - 342*n + 336)*x - (8*n^3 - 52*n^2 + 120*n - 96)*x^2 - (8*n^2 - 24*n)*x^3)*v[i - 1] + (2*n^3 - 18*n^2 + 46*n - 36 + (2*n^3 - 22*n^2 + 40*n - 12)*x - (4*n^3 - 20*n^2 + 84*n - 96)*x^2 - (16*n^2 - 32*n + 48)*x^3 - 16*n*x^4)*deriv(v[i - 1], x))/(n^3 - 7*n^2 + 14*n - 8 + (4*n^2 - 22*n + 28)*x + (4*n - 8)*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.