A053117 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	4	2							6
3	11	16	4						31
4	24	68	48	8					148
5	45	208	276	128	16				673
6	76	518	1088	928	320	32			2962
7	119	1120	3384	4480	2800	768	64		12735
8	176	2184	8928	16720	16000	7872	1792	128	53800

All 15 rows

Technical details

Node id: n-28c28f99320476cf · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cd68f890aaf2d75a

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): 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   4,    2;
   3:  11,   16,     4;
   4:  24,   68,    48,      8;
   5:  45,  208,   276,    128,     16;
   6:  76,  518,  1088,    928,    320,     32;
   7: 119, 1120,  3384,   4480,   2800,    768,     64;
   8: 176, 2184,  8928,  16720,  16000,   7872,   1792,   128;
   9: 249, 3936, 20856,  52096,  69680,  51968,  21056,  4096,  256;
  10: 340, 6666, 44352, 141856, 250432, 258720, 157696, 54272, 9216, 512;

%F Formula

The row polynomials satisfy (n^3-3*n^2+5*n-3+(n-1)*x-(n-1)*x^2)*P(n,x) = (n^3+2*n+(n^3+n)*x-n*x^2-n*x^3)*P(n-1,x) + ((2*n^2-2*n+2)*x+(n^2-n-1)*x^2-3*x^3-x^4)*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[((n^3 + 2*n + (n^3 + n)*x - n*x^2 - n*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 2)*x + (n^2 - n - 1)*x^2 - 3*x^3 - x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 5*n - 3 + (n - 1)*x - (n - 1)*x^2)]];
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] = ((n^3 + 2*n + (n^3 + n)*x - n*x^2 - n*x^3)*v[i - 1] + ((2*n^2 - 2*n + 2)*x + (n^2 - n - 1)*x^2 - 3*x^3 - x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 5*n - 3 + (n - 1)*x - (n - 1)*x^2)); 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing or…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

chebyshev_u → x ↦ x+1 → D+1

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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