A085478 → Veronese 2 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-4}(x) + \left(4\right)\,P_{n-3}(x) + \left(-5 + x\right)\,P_{n-2}(x) + \left(4\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 1\)

\(P_{3}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	1							1
3	2	1						3
4	6	5						11
5	17	17	1					35
6	45	53	9					107
7	117	163	48	1				329
8	305	495	204	13				1017
9	798	1477	774	95	1			3145
10	2090	4337	2754	523	17			9721
11	5473	12578	9391	2442	158	1		30043
12	14329	36130	31023	10274	1074	21		92851
13	37513	102974	99957	40274	6015	237	1	286971
14	98209	291550	315653	149994	29579	1921	25	886931

Technical details

Node id: n-c9095ac61976ade0 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4561f1d1eeef9015

Observed on the stored rows ?

Degrees: 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (irregular)
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:   1;
  2:   1;
  3:   2,    1;
  4:   6,    5;
  5:  17,   17,   1;
  6:  45,   53,   9;
  7: 117,  163,  48,  1;
  8: 305,  495, 204, 13;
  9: 798, 1477, 774, 95, 1;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-4,x) + 4*P(n-3,x) - (5-x)*P(n-2,x) + 4*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-P[n - 4, x] + 4*P[n - 3, x] - (5 - x)*P[n - 2, x] + 4*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; v[4] = 2 + x; for(n = 4, N, my(i = n + 1); v[i] = -v[i - 4] + 4*v[i - 3] - (5 - x)*v[i - 2] + 4*v[i - 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 (−∞, -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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from veronese_section_2_0 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] wh…?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    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 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023