A188440 → D+1

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(-2 - 6\,x + 4\,x^{2} - n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(2 - 6\,x + 8\,x^{2} - 8\,x^{3} - 7\,n + 6\,n x - 12\,n x^{2} + 8\,n x^{3} + 3\,n^{2} - 4\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 16\,x^{3} - 16\,x^{4} - 2\,n - 6\,n x - 12\,n x^{2} - 16\,n x^{3} - 2\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	3	2			5
3	3	2			5
4	5	12	4		21
5	5	12	4		21
6	7	30	36	8	81
7	7	30	36	8	81

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-08469142fb9c5873 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4f4ca2dec4972ed6

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): yes

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 3,  2;
  3: 3,  2;
  4: 5, 12,   4;
  5: 5, 12,   4;
  6: 7, 30,  36,  8;
  7: 7, 30,  36,  8;
  8: 9, 56, 120, 96, 16;
  9: 9, 56, 120, 96, 16;

%F Formula

The row polynomials satisfy (n^2-n-2+(4*n-6)*x+4*x^2)*P(n,x) = (3*n^2-7*n+2+(2*n^3-4*n^2+6*n-6)*x+(8*n^2-12*n+8)*x^2+(8*n-8)*x^3)*P(n-1,x) - (2*n+(2*n^2+6*n+4)*x+(4*n^2+12*n+12)*x^2+(16*n+16)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 7*n + 2 + (2*n^3 - 4*n^2 + 6*n - 6)*x + (8*n^2 - 12*n + 8)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (2*n + (2*n^2 + 6*n + 4)*x + (4*n^2 + 12*n + 12)*x^2 + (16*n + 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (4*n - 6)*x + 4*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 7*n + 2 + (2*n^3 - 4*n^2 + 6*n - 6)*x + (8*n^2 - 12*n + 8)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (2*n + (2*n^2 + 6*n + 4)*x + (4*n^2 + 12*n + 12)*x^2 + (16*n + 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^2 - n - 2 + (4*n - 6)*x + 4*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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-90e13dda23ca7fe5
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-90e13dda23ca7fe5
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-90e13dda23ca7fe5

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-90e13dda23ca7fe5

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b186cc46021ee4e1 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A274498 → D+1

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.