A123202 → D+1

Recurrence

\(\left(-5 + 15\,x - 8\,x^{2} + x^{3} + 2\,n - 7\,n x + 2\,n x^{2} + n^{2} x\right)\,P_{n}(x) = \left(3 - 5\,x - 16\,x^{2} + 29\,x^{3} - 14\,x^{4} + 2\,x^{5} - 5\,n + 11\,n x + 9\,n x^{2} - 23\,n x^{3} + 10\,n x^{4} - n x^{5} + 2\,n^{2} - 7\,n^{2} x - 2\,n^{2} x^{2} + 8\,n^{2} x^{3} - 2\,n^{2} x^{4} + n^{3} x - n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x + 11\,x^{2} - 21\,x^{3} + 18\,x^{4} - 7\,x^{5} + x^{6} + 2\,n x - 9\,n x^{2} + 14\,n x^{3} - 9\,n x^{4} + 2\,n x^{5} + n^{2} x^{2} - 2\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-1	-2							-3
3	-6	6	7						7
4	-30	90	-39	-34					-13
5	-168	816	-1128	292	209				21
6	-1080	7200	-16200	14100	-2505	-1546			-31
7	-7920	66960	-207000	294600	-184230	24306	13327		43
8	-65520	670320	-2628360	5145000	-5281710	2554566	-263431	-130922	-57

All 15 rows

Technical details

Node id: n-dd5e7f7fbd0b1911 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4b1e3aef476c99e5

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 -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:       -1,       -2;
   3:       -6,        6,          7;
   4:      -30,       90,        -39,        -34;
   5:     -168,      816,      -1128,        292,         209;
   6:    -1080,     7200,     -16200,      14100,       -2505,      -1546;
   7:    -7920,    66960,    -207000,     294600,     -184230,      24306,       13327;
   8:   -65520,   670320,   -2628360,    5145000,    -5281710,    2554566,     -263431,   -130922;
   9:  -604800,  7257600,  -34433280,   85518720,  -120869280,   96290880,   -37756320,   3154824,   1441729;
  10: -6168960, 84913920, -472469760, 1419949440, -2547145440, 2796208128, -1811236896, 594890568, -41368977, -17572114;

%F Formula

The row polynomials satisfy (2*n-5+(n^2-7*n+15)*x+(2*n-8)*x^2+x^3)*P(n,x) = (2*n^2-5*n+3+(n^3-7*n^2+11*n-5)*x-(2*n^2-9*n+16)*x^2-(n^3-8*n^2+23*n-29)*x^3-(2*n^2-10*n+14)*x^4-(n-2)*x^5)*P(n-1,x) + ((2*n-2)*x+(n^2-9*n+11)*x^2-(2*n^2-14*n+21)*x^3+(n^2-9*n+18)*x^4+(2*n-7)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 + (n^3 - 7*n^2 + 11*n - 5)*x - (2*n^2 - 9*n + 16)*x^2 - (n^3 - 8*n^2 + 23*n - 29)*x^3 - (2*n^2 - 10*n + 14)*x^4 - (n - 2)*x^5)*P[n - 1, x] + ((2*n - 2)*x + (n^2 - 9*n + 11)*x^2 - (2*n^2 - 14*n + 21)*x^3 + (n^2 - 9*n + 18)*x^4 + (2*n - 7)*x^5 + x^6)*D[P[n - 1, x], x])/(2*n - 5 + (n^2 - 7*n + 15)*x + (2*n - 8)*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 + (n^3 - 7*n^2 + 11*n - 5)*x - (2*n^2 - 9*n + 16)*x^2 - (n^3 - 8*n^2 + 23*n - 29)*x^3 - (2*n^2 - 10*n + 14)*x^4 - (n - 2)*x^5)*v[i - 1] + ((2*n - 2)*x + (n^2 - 9*n + 11)*x^2 - (2*n^2 - 14*n + 21)*x^3 + (n^2 - 9*n + 18)*x^4 + (2*n - 7)*x^5 + x^6)*deriv(v[i - 1], x))/(2*n - 5 + (n^2 - 7*n + 15)*x + (2*n - 8)*x^2 + x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023