A091042 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-39 + 39\,x - 6\,x^{2} + 53\,n - 38\,n x + 4\,n x^{2} - 24\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-6 - 67\,x + 119\,x^{2} - 22\,x^{3} + 17\,n + 97\,n x - 106\,n x^{2} + 12\,n x^{3} - 12\,n^{2} - 60\,n^{2} x + 24\,n^{2} x^{2} + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(22\,x - 58\,x^{2} + 44\,x^{3} - 8\,x^{4} - 24\,n x + 40\,n x^{2} - 16\,n x^{3} + 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	1														5
3	15	12	1													28
4	42	77	24	1												144
5	93	336	234	40	1											704
6	176	1089	1452	550	60	1										3328
7	299	2860	6435	4576	1105	84	1									15360
8	470	6461	22308	26455	11830	1995	112	1								69632
9	697	13056	64532	116688	86190	26656	3332	144	1							311296
10	988	24225	162792	419900	470288	238374	54264	5244	180	1						1376256
11	1351	42028	369189	1292000	2057510	1573656	583338	102144	7875	220	1					6029312
12	1794	69069	769120	3513917	7577580	8216474	4576264	1297890	180642	11385	264	1				26214400
13	2325	108560	1495230	8652600	24329975	35659200	28081620	11921360	2675475	303600	15950	312	1			113246208
14	2952	164385	2744820	19635330	69876300	133387695	141745320	84874140	28416960	5180175	489060	21762	364	1		486539264
15	3683	241164	4801095	41620800	183001455	441780780	610775235	492726240	232166025	62951460	9517365	760032	29029	420	1	2080374784

Technical details

Node id: n-abac71f8b01d0ff6 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a95c31fc8093635f

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,     1;
   3:  15,    12,      1;
   4:  42,    77,     24,      1;
   5:  93,   336,    234,     40,      1;
   6: 176,  1089,   1452,    550,     60,      1;
   7: 299,  2860,   6435,   4576,   1105,     84,     1;
   8: 470,  6461,  22308,  26455,  11830,   1995,   112,    1;
   9: 697, 13056,  64532, 116688,  86190,  26656,  3332,  144,   1;
  10: 988, 24225, 162792, 419900, 470288, 238374, 54264, 5244, 180, 1;

%F Formula

The row polynomials satisfy (4*n^3-24*n^2+53*n-39+(8*n^2-38*n+39)*x+(4*n-6)*x^2)*P(n,x) = (4*n^3-12*n^2+17*n-6+(12*n^3-60*n^2+97*n-67)*x+(24*n^2-106*n+119)*x^2+(12*n-22)*x^3)*P(n-1,x) + ((8*n^2-24*n+22)*x-(8*n^2-40*n+58)*x^2-(16*n-44)*x^3-8*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[((4*n^3 - 12*n^2 + 17*n - 6 + (12*n^3 - 60*n^2 + 97*n - 67)*x + (24*n^2 - 106*n + 119)*x^2 + (12*n - 22)*x^3)*P[n - 1, x] + ((8*n^2 - 24*n + 22)*x - (8*n^2 - 40*n + 58)*x^2 - (16*n - 44)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 24*n^2 + 53*n - 39 + (8*n^2 - 38*n + 39)*x + (4*n - 6)*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] = ((4*n^3 - 12*n^2 + 17*n - 6 + (12*n^3 - 60*n^2 + 97*n - 67)*x + (24*n^2 - 106*n + 119)*x^2 + (12*n - 22)*x^3)*v[i - 1] + ((8*n^2 - 24*n + 22)*x - (8*n^2 - 40*n + 58)*x^2 - (16*n - 44)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 24*n^2 + 53*n - 39 + (8*n^2 - 38*n + 39)*x + (4*n - 6)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from reversal of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    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 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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