A330381 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(9 + 3\,x - 9\,x^{2}\right)\,P_{n-3}(x) + \left(1 - 3\,x - 3\,x^{2} - 4\,x^{3}\right)\,P_{n-3}'(x) + \left(-22 - 2\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(-2 + 6\,x + 8\,x^{2}\right)\,P_{n-2}'(x) + \left(15 + 8\,x\right)\,P_{n-1}(x) + \left(1 - 4\,x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

\(P_{2}(x) = 6 + 11\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	6	11	3						20
3	10	35	25	4					74
4	15	84	113	46	5				263
5	21	170	363	276	75	6			911
6	28	307	941	1131	571	113	7		3098
7	36	511	2107	3623	2885	1057	161	8	10388

All 15 rows

Technical details

Node id: n-50e231117764dacd · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d39cf0ea889e3608

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:  3,    2;
  2:  6,   11,    3;
  3: 10,   35,   25,     4;
  4: 15,   84,  113,    46,     5;
  5: 21,  170,  363,   276,    75,     6;
  6: 28,  307,  941,  1131,   571,   113,     7;
  7: 36,  511, 2107,  3623,  2885,  1057,   161,    8;
  8: 45,  800, 4242,  9780, 11123,  6420,  1806,  220,   9;
  9: 55, 1194, 7878, 23280, 35441, 29194, 12930, 2904, 291, 10;

%F Formula

The row polynomials satisfy 4*P(n,x) = (9+3*x-9*x^2)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (22+2*x+4*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (15+8*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 6 + 11*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 6 + 11*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 + 3*x - 9*x^2)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (22 + 2*x + 4*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (15 + 8*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];
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] = 3 + 2*x; v[3] = 6 + 11*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((9 + 3*x - 9*x^2)*v[i - 3] + (1 - 3*x - 3*x^2 - 4*x^3)*deriv(v[i - 3], x) - (22 + 2*x + 4*x^2)*v[i - 2] - (2 - 6*x - 8*x^2)*deriv(v[i - 2], x) + (15 + 8*x)*v[i - 1] + (1 - 4*x)*deriv(v[i - 1], x))/(4)); 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 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
  - 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, read by rows, of pairwise sums of trinomial coefficients (A027907).?
    
    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 read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    ↑ 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.