Veronese parent of A098172 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	x²⁹	x³⁰	x³¹	x³²	x³³	x³⁴	x³⁵	x³⁶	x³⁷	x³⁸	x³⁹	x⁴⁰	x⁴¹	x⁴²	x⁴³	P_n(1)
1	2																																											2
2	0	0	4	5																																								9
3	0	0	0	0	6	14	8																																					28
4	0	0	0	0	0	0	8	27	30	11																																		76
5	0	0	0	0	0	0	0	0	10	44	72	52	14																															192
6	0	0	0	0	0	0	0	0	0	0	12	65	140	150	80	17																												464
7	0	0	0	0	0	0	0	0	0	0	0	0	14	90	240	340	270	114	20																									1088
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	16	119	378	665	700	441	154	23																						2496
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	18	152	560	1176	1540	1288	672	200	26																			5632
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	20	189	792	1932	3024	3150	2184	972	252	29																12544
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	22	230	1080	3000	5460	6804	5880	3480	1350	310	32													27648
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	24	275	1430	4455	9240	13398	13860	10230	5280	1815	374	35										60416
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	26	324	1848	6380	14850	24552	29568	26136	16830	7700	2376	444	38							131072
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	28	377	2340	8866	22880	42471	58344	60060	46332	26455	10868	3042	520	41				282624
15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	30	434	2912	12012	34034	70070	108108	126984	114114	78078	40040	14924	3822	602	44	606208

Technical details

Node id: n-95dbcd414f8b4440 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b3b9dd9fef169193

Observed on the stored rows ?

Degrees: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43 (affine: slope 3, intercept -2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 0, 2;
   2: 0, 0, 0, 4, 5;
   3: 0, 0, 0, 0, 0, 6, 14, 8;
   4: 0, 0, 0, 0, 0, 0,  0, 8, 27, 30, 11;
   5: 0, 0, 0, 0, 0, 0,  0, 0,  0, 10, 44, 72, 52,  14;
   6: 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 12, 65, 140, 150,  80,  17;
   7: 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  14,  90, 240, 340, 270, 114,  20;
   8: 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  16, 119, 378, 665, 700,  441,  154,   23;
   9: 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  18, 152, 560, 1176, 1540, 1288,  672,  200,   26;
  10: 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  20,  189,  792, 1932, 3024, 3150, 2184, 972, 252, 29;

%F Formula

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

%t Mathematica

P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n*x^2 + (5*n - 1)*x^3 + (3*n - 1)*x^4)*P[n - 1, x])/(2*n - 2 + (3*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2*x; for(n = 2, N, my(i = n); v[i] = ((2*n*x^2 + (5*n - 1)*x^3 + (3*n - 1)*x^4)*v[i - 1])/(2*n - 2 + (3*n - 4)*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, Laguerre’s theorem on multiplier sequences
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, Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators, Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.