A358735 → basis A007318 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	2	8	7						17
3	6	34	60	33					133
4	24	172	442	484	191				1313
5	120	1028	3408	5480	4283	1304			15623
6	720	7128	28648	59972	69096	41604	10241		217409
7	5040	56376	264568	676608	1020012	907612	441840	90865	3462921

All 15 rows

Technical details

Node id: n-f325e9d4bddfdf8c · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-cfc761a86542e28e

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:      1,       2;
  2:      2,       8,        7;
  3:      6,      34,       60,        33;
  4:     24,     172,      442,       484,       191;
  5:    120,    1028,     3408,      5480,      4283,      1304;
  6:    720,    7128,    28648,     59972,     69096,     41604,     10241;
  7:   5040,   56376,   264568,    676608,   1020012,    907612,    441840,     90865;
  8:  40320,  501408,  2679584,   8051416,  14897528,  17401044,  12541744,   5103716,   898409;
  9: 362880, 4956192, 29626704, 101881792, 222378760, 319805596, 303267168, 182985016, 63784717, 9791634;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) + (n+(n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] + (n + (n + 2)*x)*P[n - 1, 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] + (n + (n + 2)*x)*v[i - 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 (-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

A358735 → basis A007318

References

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