A188440 → θ+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

\(\left(2 + 3\,x + 2\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 - 5\,x - 12\,x^{2} - 4\,x^{3} + 10\,n x + 5\,n x^{2} - 2\,n x^{3} + 9\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 16\,x^{2} - 20\,x^{3} - 8\,x^{4} - 6\,n x^{2} - 18\,n x^{3} - 12\,n x^{4} - 2\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	1	4			5
3	1	4			5
4	1	8	12		21
5	1	8	12		21
6	1	12	36	32	81
7	1	12	36	32	81

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-5265efb6991612b1 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4a9146a6908256f6

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: 1,  4;
  3: 1,  4;
  4: 1,  8, 12;
  5: 1,  8, 12;
  6: 1, 12, 36,  32;
  7: 1, 12, 36,  32;
  8: 1, 16, 72, 128, 80;
  9: 1, 16, 72, 128, 80;

%F Formula

The row polynomials satisfy (2+(2*n+3)*x+(n^2+n)*x^2)*P(n,x) = (2+(10*n-5)*x+(9*n^2+5*n-12)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (4*x+(6*n+16)*x^2+(2*n^2+18*n+20)*x^3+(4*n^2+12*n+8)*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[((2 + (10*n - 5)*x + (9*n^2 + 5*n - 12)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (4*x + (6*n + 16)*x^2 + (2*n^2 + 18*n + 20)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 3)*x + (n^2 + n)*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] = ((2 + (10*n - 5)*x + (9*n^2 + 5*n - 12)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (4*x + (6*n + 16)*x^2 + (2*n^2 + 18*n + 20)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 3)*x + (n^2 + n)*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-f85b46c82d805226
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-f85b46c82d805226
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-f85b46c82d805226

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 euler_polynomial, shift 0 e-f85b46c82d805226

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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 → θ+1

References

[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.