A082985 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−5, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	11	7	1													19
3	29	31	10	1												71
4	76	115	60	13	1											265
5	199	390	285	98	16	1										989
6	521	1254	1185	566	145	19	1									3691
7	1364	3893	4524	2785	985	201	22	1								13775
8	3571	11789	16280	12313	5595	1569	266	25	1							51409
9	9349	35045	56105	50434	28113	10101	2345	340	28	1						191861
10	24476	102695	187080	195094	129178	56847	16870	3340	423	31	1					716035
11	64079	297516	607830	721928	554515	289618	105112	26550	4581	515	34	1				2672279
12	167761	853932	1933926	2578520	2256295	1366522	588084	181422	39870	6095	616	37	1			9973081
13	439204	2432041	6047880	8947558	8792890	6066243	3025662	1105800	296451	57640	7909	726	40	1		37220045
14	1149851	6881395	18641755	30312034	33069933	25625097	14555145	6161640	1955283	463276	80751	10050	845	43	1	138907099

Technical details

Node id: n-02870b15db72fb9a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
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 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:    4,     1;
  2:   11,     7,     1;
  3:   29,    31,    10,     1;
  4:   76,   115,    60,    13,     1;
  5:  199,   390,   285,    98,    16,     1;
  6:  521,  1254,  1185,   566,   145,    19,    1;
  7: 1364,  3893,  4524,  2785,   985,   201,   22,   1;
  8: 3571, 11789, 16280, 12313,  5595,  1569,  266,  25,  1;
  9: 9349, 35045, 56105, 50434, 28113, 10101, 2345, 340, 28, 1;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = (3*n+1+n*x)*P(n-1,x) + (5+6*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 1 + n*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + 1 + n*x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(2*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 (-5, -1)
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from shift_x_plus_one of Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

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