A082985 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 + 2\,x + x^{2} + 13\,n - 4\,n x - 2\,n x^{2} - 3\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(6 - 2\,x - x^{2} + 13\,n - 2\,n x - 5\,n x^{2} - n x^{3} + 3\,n^{2} + n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 14\,x^{2} - 8\,x^{3} - x^{4} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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	10	7	1													18
3	21	28	10	1												60
4	39	84	54	13	1											191
5	66	209	209	88	16	1										589
6	104	455	650	416	130	19	1									1775
7	155	896	1728	1550	725	180	22	1								5257
8	221	1632	4080	4862	3145	1156	238	25	1							15360
9	304	2793	8778	13376	11362	5719	1729	304	28	1						44394
10	406	4543	17523	33176	35672	23373	9604	2464	378	31	1					127171
11	529	7084	32890	75647	100165	82432	43792	15180	3381	460	34	1				361595
12	675	10660	58630	160875	256750	258400	171360	76380	22875	4500	550	37	1			1021693
13	846	15561	100035	322608	609960	735318	593028	328320	125874	33165	5841	648	40	1		2871245
14	1044	22127	164372	615264	1358708	1929602	1854666	1243056	589512	198099	46574	7424	754	43	1	8031246

Technical details

Node id: n-24c7e31c40ab68b4 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5aabb12a2c39c57b

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:  10,    7,    1;
  3:  21,   28,   10,     1;
  4:  39,   84,   54,    13,     1;
  5:  66,  209,  209,    88,    16,    1;
  6: 104,  455,  650,   416,   130,   19,    1;
  7: 155,  896, 1728,  1550,   725,  180,   22,   1;
  8: 221, 1632, 4080,  4862,  3145, 1156,  238,  25,  1;
  9: 304, 2793, 8778, 13376, 11362, 5719, 1729, 304, 28, 1;

%F Formula

The row polynomials satisfy (2*n^3-3*n^2+13*n-6-(4*n-2)*x-(2*n-1)*x^2)*P(n,x) = (2*n^3+3*n^2+13*n+6+(n^3+n^2-2*n-2)*x-(5*n+1)*x^2-n*x^3)*P(n-1,x) + ((4*n^2+8)*x+(n^2-14)*x^2-8*x^3-x^4)*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[((2*n^3 + 3*n^2 + 13*n + 6 + (n^3 + n^2 - 2*n - 2)*x - (5*n + 1)*x^2 - n*x^3)*P[n - 1, x] + ((4*n^2 + 8)*x + (n^2 - 14)*x^2 - 8*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 13*n - 6 - (4*n - 2)*x - (2*n - 1)*x^2)]];
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] = ((2*n^3 + 3*n^2 + 13*n + 6 + (n^3 + n^2 - 2*n - 2)*x - (5*n + 1)*x^2 - n*x^3)*v[i - 1] + ((4*n^2 + 8)*x + (n^2 - 14)*x^2 - 8*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 13*n - 6 - (4*n - 2)*x - (2*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 - 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 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 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    
    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
    
    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 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    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
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A111125 → D+1

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