A080248 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	8	3						12
3	1	26	30	4					61
4	1	80	219	80	5				385
5	1	242	1434	1092	175	6			2950
6	1	728	8967	12832	3990	336	7		26861
7	1	2186	54894	140276	75890	11844	588	8	285687

All 15 rows

Technical details

Node id: n-2def0b7e202332d2 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-21ed105e22eabefe

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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;
  2: 1,     8,       3;
  3: 1,    26,      30,        4;
  4: 1,    80,     219,       80,        5;
  5: 1,   242,    1434,     1092,      175,       6;
  6: 1,   728,    8967,    12832,     3990,     336,       7;
  7: 1,  2186,   54894,   140276,    75890,   11844,     588,     8;
  8: 1,  6560,  332643,  1475952,  1313695,  339792,   30282,   960,    9;
  9: 1, 19682, 2005698, 15203044, 21550365, 8712066, 1244320, 69168, 1485, 10;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-2*x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-2*x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x))/(2)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Stirling-like number triangle defined by sequence A000217.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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.