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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	P_n(1)
1																						0
2	0	2	3																			5
3	0	0	0	4	10	6																20
4	0	0	0	0	0	6	21	24	9													60
5	0	0	0	0	0	0	0	8	36	60	44	12										160
6	0	0	0	0	0	0	0	0	0	10	55	120	130	70	15							400
7	0	0	0	0	0	0	0	0	0	0	0	12	78	210	300	240	102	18				960
8	0	0	0	0	0	0	0	0	0	0	0	0	0	14	105	336	595	630	399	140	21	2240

All 15 rows

Technical details

Node id: n-2224fadf43dd48ee · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8a8f26bfa265b5f7

Observed on the stored rows ?

Degrees: -1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41 (affine: slope 3, intercept -4)
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: 0, 2, 3;
   3: 0, 0, 0, 4, 10, 6;
   4: 0, 0, 0, 0,  0, 6, 21, 24,  9;
   5: 0, 0, 0, 0,  0, 0,  0,  8, 36, 60, 44,  12;
   6: 0, 0, 0, 0,  0, 0,  0,  0,  0, 10, 55, 120, 130,  70,  15;
   7: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  12,  78, 210, 300, 240, 102,  18;
   8: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  14, 105, 336, 595, 630,  399,  140,   21;
   9: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  16, 136, 504, 1064, 1400, 1176,  616,  184,   24;
  10: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  18,  171,  720, 1764, 2772, 2898, 2016, 900, 234, 27;

%F Formula

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

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n - 1)*x^2 + (n - 1)*x^3)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (((n - 1)*x^2 + (n - 1)*x^3)*v[i - 1])/(n - 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

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).?
  - 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
  - Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

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.