A257620 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(18 - 81\,x + 126\,x^{2} - 81\,x^{3} + 18\,x^{4} - 18\,n + 63\,n x - 108\,n x^{2} + 90\,n x^{3} - 18\,n x^{4} - 18\,n^{2} x + 45\,n^{2} x^{2} - 27\,n^{2} x^{3} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(9\,x - 45\,x^{2} + 81\,x^{3} - 63\,x^{4} + 18\,x^{5} - 9\,n x + 36\,n x^{2} - 54\,n x^{3} + 27\,n x^{4} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-15 + 36\,x - 27\,x^{2} + 6\,x^{3} + 9\,n - 21\,n x + 15\,n x^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 12\,x^{2} - 6\,x^{3} + 3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	3							9
2	45	54	9						108
3	324	891	378	27					1620
4	2187	12798	11664	2430	81				29160
5	14094	160623	293544	128790	15066	243			612360
6	88209	1823958	6152031	5234220	1305639	91854	729		14696640
7	542376	19317771	111865050	170793765	81102708	12629925	555498	2187	396809280

All 15 rows

Technical details

Node id: n-08bb072bfddb6a12 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d762c6db5469cb8f

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:       45,         54,           9;
  3:      324,        891,         378,           27;
  4:     2187,      12798,       11664,         2430,           81;
  5:    14094,     160623,      293544,       128790,        15066,         243;
  6:    88209,    1823958,     6152031,      5234220,      1305639,       91854,         729;
  7:   542376,   19317771,   111865050,    170793765,     81102708,    12629925,      555498,       2187;
  8:  3300183,  194979798,  1832552910,   4677953634,   3919278960,  1153961802,   118898442,    3346110,     6561;
  9: 19958562, 1903208319, 27820267128, 112126335264, 154750186692, 79548962598, 15550987176, 1101145752, 20116026, 19683;

%F Formula

The row polynomials satisfy (n-2+2*x)*P(n,x) = -(18*n-18+(18*n^2-63*n+81)*x+(9*n^3-45*n^2+108*n-126)*x^2+(27*n^2-90*n+81)*x^3+(18*n-18)*x^4)*P(n-2,x) - ((9*n-9)*x+(9*n^2-36*n+45)*x^2-(9*n^2-54*n+81)*x^3-(27*n-63)*x^4-18*x^5)*d/dx P(n-2,x) + (9*n-15+(6*n^2-21*n+36)*x+(15*n-27)*x^2+6*x^3)*P(n-1,x) + ((3*n-6)*x-(3*n-12)*x^2-6*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[(-(18*n - 18 + (18*n^2 - 63*n + 81)*x + (9*n^3 - 45*n^2 + 108*n - 126)*x^2 + (27*n^2 - 90*n + 81)*x^3 + (18*n - 18)*x^4)*P[n - 2, x] - ((9*n - 9)*x + (9*n^2 - 36*n + 45)*x^2 - (9*n^2 - 54*n + 81)*x^3 - (27*n - 63)*x^4 - 18*x^5)*D[P[n - 2, x], x] + (9*n - 15 + (6*n^2 - 21*n + 36)*x + (15*n - 27)*x^2 + 6*x^3)*P[n - 1, x] + ((3*n - 6)*x - (3*n - 12)*x^2 - 6*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*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] = (-(18*n - 18 + (18*n^2 - 63*n + 81)*x + (9*n^3 - 45*n^2 + 108*n - 126)*x^2 + (27*n^2 - 90*n + 81)*x^3 + (18*n - 18)*x^4)*v[i - 2] - ((9*n - 9)*x + (9*n^2 - 36*n + 45)*x^2 - (9*n^2 - 54*n + 81)*x^3 - (27*n - 63)*x^4 - 18*x^5)*deriv(v[i - 2], x) + (9*n - 15 + (6*n^2 - 21*n + 36)*x + (15*n - 27)*x^2 + 6*x^3)*v[i - 1] + ((3*n - 6)*x - (3*n - 12)*x^2 - 6*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? 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(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

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.