A091042 → Veronese 2 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(191 - 381\,x + 189\,x^{2} + x^{3}\right)\,P_{n-6}(x) + \left(-64 + 192\,x - 192\,x^{2} + 64\,x^{3}\right)\,P_{n-6}'(x) + \left(3342 - 2252\,x - 66\,x^{2}\right)\,P_{n-5}(x) + \left(-128 + 1264\,x - 1120\,x^{2} - 16\,x^{3}\right)\,P_{n-5}'(x) + \left(9041 + 146\,x - 3\,x^{2}\right)\,P_{n-4}(x) + \left(1152 + 5184\,x + 64\,x^{2}\right)\,P_{n-4}'(x) + \left(1860 + 52\,x\right)\,P_{n-3}(x) + \left(-2048 + 416\,x + 32\,x^{2}\right)\,P_{n-3}'(x) + \left(-111 + 3\,x\right)\,P_{n-2}(x) + \left(1472 + 128\,x\right)\,P_{n-2}'(x) + \left(14\right)\,P_{n-1}(x) + \left(-384 - 16\,x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 0\)

\(P_{3}(x) = 5\)

\(P_{4}(x) = 35\)

\(P_{5}(x) = 126 + 18\,x\)

\(P_{6}(x) = 330 + 330\,x\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
1				0
2				0
3	5			5
4	35			35
5	126	18		144
6	330	330		660
7	715	2574	39	3328
8	1365	12870	1365	15600

All 15 rows

Technical details

Node id: n-e066b43456aa872f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-53291945d5eef4b9

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -3)
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:    0;
   2:    0;
   3:    5;
   4:   35;
   5:  126,     18;
   6:  330,    330;
   7:  715,   2574,     39;
   8: 1365,  12870,   1365;
   9: 2380,  48620,  18564,   68;
  10: 3876, 151164, 151164, 3876;

%F Formula

The row polynomials satisfy P(n,x) = (191-381*x+189*x^2+x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3342-2252*x-66*x^2)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (9041+146*x-3*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) + (1860+52*x)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) - (111-3*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) + 14*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 5, P(4,x) = 35, P(5,x) = 126 + 18*x, P(6,x) = 330 + 330*x.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 5;
P[4, x_] := 35;
P[5, x_] := 126 + 18*x;
P[6, x_] := 330 + 330*x;
P[n_, x_] := P[n, x] = Expand[(191 - 381*x + 189*x^2 + x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3342 - 2252*x - 66*x^2)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (9041 + 146*x - 3*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] + (1860 + 52*x)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] - (111 - 3*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] + 14*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 5; v[4] = 35; v[5] = 126 + 18*x; v[6] = 330 + 330*x; for(n = 7, N, my(i = n); v[i] = (191 - 381*x + 189*x^2 + x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3342 - 2252*x - 66*x^2)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (9041 + 146*x - 3*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) + (1860 + 52*x)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) - (111 - 3*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) + 14*v[i - 1] - (384 + 16*x)*deriv(v[i - 1], x)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from veronese_section_2_0 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] 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) 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)

Related ?

Same fingerprint ?

A091044 → Veronese 2 (j=1)

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[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