A111125 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-3 - 14\,x - 10\,x^{2} + 2\,n + 9\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	1	7	11													19
4	1	10	31	29												71
5	1	13	60	115	76											265
6	1	16	98	285	390	199										989
7	1	19	145	566	1185	1254	521									3691
8	1	22	201	985	2785	4524	3893	1364								13775
9	1	25	266	1569	5595	12313	16280	11789	3571							51409
10	1	28	340	2345	10101	28113	50434	56105	35045	9349						191861
11	1	31	423	3340	16870	56847	129178	195094	187080	102695	24476					716035
12	1	34	515	4581	26550	105112	289618	554515	721928	607830	297516	64079				2672279
13	1	37	616	6095	39870	181422	588084	1366522	2256295	2578520	1933926	853932	167761			9973081
14	1	40	726	7909	57640	296451	1105800	3025662	6066243	8792890	8947558	6047880	2432041	439204		37220045
15	1	43	845	10050	80751	463276	1955283	6161640	14555145	25625097	33069933	30312034	18641755	6881395	1149851	138907099

Technical details

Node id: n-9436d64c250aa083 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-2c11568138d58d94

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: 1,  4;
   3: 1,  7,  11;
   4: 1, 10,  31,   29;
   5: 1, 13,  60,  115,    76;
   6: 1, 16,  98,  285,   390,   199;
   7: 1, 19, 145,  566,  1185,  1254,   521;
   8: 1, 22, 201,  985,  2785,  4524,  3893,  1364;
   9: 1, 25, 266, 1569,  5595, 12313, 16280, 11789,  3571;
  10: 1, 28, 340, 2345, 10101, 28113, 50434, 56105, 35045, 9349;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = (2*n-3+(9*n-14)*x+(5*n-10)*x^2)*P(n-1,x) - (x+6*x^2+5*x^3)*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 - 3 + (9*n - 14)*x + (5*n - 10)*x^2)*P[n - 1, x] - (x + 6*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 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 - 3 + (9*n - 14)*x + (5*n - 10)*x^2)*v[i - 1] - (x + 6*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 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 (-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ Three-term recurrence criterion (above)

Related ?

Candidate equalities ?

A111125 → reverse → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

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