Veronese parent of A109447 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1	2							3
1	2	2							4
2	1	6	9	4					20
3	4	12	12	4					32
4	1	10	30	40	25	6			112
5	6	30	60	60	30	6			192
6	1	14	63	140	175	126	49	8	576
7	8	56	168	280	280	168	56	8	1024

All 15 rows

Technical details

Node id: n-8efa9190cdce6a49 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-bc2c1175f3d1ff4b

Observed on the stored rows ?

Degrees: 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1,  2;
  1:  2,  2;
  2:  1,  6,   9,   4;
  3:  4, 12,  12,   4;
  4:  1, 10,  30,  40,   25,    6;
  5:  6, 30,  60,  60,   30,    6;
  6:  1, 14,  63, 140,  175,  126,  49,   8;
  7:  8, 56, 168, 280,  280,  168,  56,   8;
  8:  1, 18, 108, 336,  630,  756, 588, 288, 81, 10;
  9: 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 - 3*n - (3*n^2 + 5*n)*x - (n^3 + 3*n^2 + 2*n)*x^2)*v[i - 1] + (n^2 + n - 1 + (n^2 - n - 4)*x - (n^2 + 5*n + 5)*x^2 - (n^2 + 3*n + 2)*x^3)*deriv(v[i - 1], x))/(n^2 + n)); 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 veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? 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.