A116414 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n - 22\,n x - 16\,n x^{2} + 8\,n x^{3} - 20\,n^{2} + 8\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{3} + 8\,n^{3} x\right)\,P_{n}(x) = \left(40\,x^{2} + 28\,x^{3} - 16\,x^{4} - 16\,x^{5} - 2\,n - 58\,n x - 184\,n x^{2} - 116\,n x^{3} + 16\,n x^{4} + 16\,n x^{5} - 4\,n^{2} - 68\,n^{2} x + 40\,n^{2} x^{2} + 112\,n^{2} x^{3} + 32\,n^{2} x^{4} + 16\,n^{3} + 104\,n^{3} x + 80\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 20\,x^{3} - 24\,x^{4} - 48\,x^{5} - 16\,x^{6} - 2\,n x - 28\,n x^{2} - 112\,n x^{3} - 112\,n x^{4} - 32\,n x^{5} - 8\,n^{2} x - 68\,n^{2} x^{2} - 64\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	4							9
2	9	34	13						56
3	13	96	162	40					311
4	17	190	639	668	121				1635
5	21	316	1636	3420	2551	364			8308
6	25	474	3345	10856	16134	9294	1093		41221
7	29	664	5958	26560	61490	70128	32804	3280	200913

All 15 rows

Technical details

Node id: n-640e334a90295685 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-602803251b9c8a8c

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:  5,    4;
  2:  9,   34,    13;
  3: 13,   96,   162,     40;
  4: 17,  190,   639,    668,    121;
  5: 21,  316,  1636,   3420,   2551,     364;
  6: 25,  474,  3345,  10856,  16134,    9294,    1093;
  7: 29,  664,  5958,  26560,  61490,   70128,   32804,    3280;
  8: 33,  886,  9667,  55140, 176135,  313052,  287874,  113176,   9841;
  9: 37, 1140, 14664, 102228, 419601, 1031388, 1476180, 1133016, 383817, 29524;

%F Formula

The row polynomials satisfy (16*n^3-20*n^2+6*n+(8*n^3+8*n^2-22*n)*x+(16*n^2-16*n)*x^2+8*n*x^3)*P(n,x) = (16*n^3-4*n^2-2*n+(104*n^3-68*n^2-58*n)*x+(80*n^3+40*n^2-184*n+40)*x^2+(16*n^3+112*n^2-116*n+28)*x^3+(32*n^2+16*n-16)*x^4+(16*n-16)*x^5)*P(n-1,x) - ((8*n^2+2*n)*x+(68*n^2+28*n-3)*x^2+(64*n^2+112*n-20)*x^3+(16*n^2+112*n+24)*x^4+(32*n+48)*x^5+16*x^6)*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[((16*n^3 - 4*n^2 - 2*n + (104*n^3 - 68*n^2 - 58*n)*x + (80*n^3 + 40*n^2 - 184*n + 40)*x^2 + (16*n^3 + 112*n^2 - 116*n + 28)*x^3 + (32*n^2 + 16*n - 16)*x^4 + (16*n - 16)*x^5)*P[n - 1, x] - ((8*n^2 + 2*n)*x + (68*n^2 + 28*n - 3)*x^2 + (64*n^2 + 112*n - 20)*x^3 + (16*n^2 + 112*n + 24)*x^4 + (32*n + 48)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(16*n^3 - 20*n^2 + 6*n + (8*n^3 + 8*n^2 - 22*n)*x + (16*n^2 - 16*n)*x^2 + 8*n*x^3)]];
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] = ((16*n^3 - 4*n^2 - 2*n + (104*n^3 - 68*n^2 - 58*n)*x + (80*n^3 + 40*n^2 - 184*n + 40)*x^2 + (16*n^3 + 112*n^2 - 116*n + 28)*x^3 + (32*n^2 + 16*n - 16)*x^4 + (16*n - 16)*x^5)*v[i - 1] - ((8*n^2 + 2*n)*x + (68*n^2 + 28*n - 3)*x^2 + (64*n^2 + 112*n - 20)*x^3 + (16*n^2 + 112*n + 24)*x^4 + (32*n + 48)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(16*n^3 - 20*n^2 + 6*n + (8*n^3 + 8*n^2 - 22*n)*x + (16*n^2 - 16*n)*x^2 + 8*n*x^3)); 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 Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).?
    
    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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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