A154233 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(x^{2} - n x^{2} - 5\,n^{2} x^{2} + n^{3} x^{2} + 3\,n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-4}(x) + \left(2\,x + 4\,n x + 2\,n x^{2} + 2\,n^{2} x^{2} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(-1 + x^{2} - n - 6\,n x - n x^{2} + 2\,n^{3} x\right)\,P_{n-2}(x) + \left(-2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 7 + 2\,x\)

\(P_{3}(x) = 19 + 38\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	7	2						9
3	19	38	3					60
4	39	342	117	4				502
5	69	1522	2283	276	5			4155
6	111	4858	25599	9716	555	6		40845
7	167	12670	158451	211268	31675	1002	7	415240

All 15 rows

Technical details

Node id: n-f56a394412c7f59c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a02b64eb9cb9fbe9

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:   7,     2;
  3:  19,    38,       3;
  4:  39,   342,     117,        4;
  5:  69,  1522,    2283,      276,        5;
  6: 111,  4858,   25599,     9716,      555,       6;
  7: 167, 12670,  158451,   211268,    31675,    1002,      7;
  8: 239, 28766,  694833,  2845908,  1158055,   86298,   1673,    8;
  9: 329, 58970, 2429427, 22575204, 28219005, 4858854, 206395, 2632, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(n^5-3*n^4-n^3+5*n^2+n-1)*x^2*P(n-4,x) - ((2*n^3-4*n-2)*x+(2*n^3-2*n^2-2*n)*x^2)*P(n-3,x) - (n+1-(2*n^3-6*n)*x+(n-1)*x^2)*P(n-2,x) + (2*n+(2*n-2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 7 + 2*x, P(3,x) = 19 + 38*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 7 + 2*x;
P[3, x_] := 19 + 38*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*P[n - 4, x] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*P[n - 3, x] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*P[n - 2, x] + (2*n + (2*n - 2)*x)*P[n - 1, x])/(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] = 7 + 2*x; v[4] = 19 + 38*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*v[i - 4] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*v[i - 3] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*v[i - 2] + (2*n + (2*n - 2)*x)*v[i - 1])/(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) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.