A086645 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x - 3\,n + 13\,n x - 9\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(6 + 30\,x + 4\,x^{2} - 3\,n - 5\,n x + 12\,n x^{2} - 9\,n^{2} x - 19\,n^{2} x^{2} + 2\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x + 4\,x^{2} + 2\,x^{3} - 4\,n x^{2} + 4\,n x^{3} + 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	2							3
3	1	12	3						16
4	1	30	45	4					80
5	1	56	210	112	5				384
6	1	90	630	840	225	6			1792
7	1	132	1485	3696	2475	396	7		8192
8	1	182	3003	12012	15015	6006	637	8	36864

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b7536fd26109db74 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5953f24a720dd506

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 -1)
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

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,  12,    3;
   4: 1,  30,   45,     4;
   5: 1,  56,  210,   112,      5;
   6: 1,  90,  630,   840,    225,      6;
   7: 1, 132, 1485,  3696,   2475,    396,      7;
   8: 1, 182, 3003, 12012,  15015,   6006,    637,     8;
   9: 1, 240, 5460, 32032,  64350,  48048,  12740,   960,    9;
  10: 1, 306, 9180, 74256, 218790, 262548, 129948, 24480, 1377, 10;

%F Formula

The row polynomials satisfy -(3*n-6-(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = -(3*n-6-(2*n^3-9*n^2-5*n+30)*x-(6*n^3-19*n^2+12*n+4)*x^2)*P(n-1,x) - (6*x-(4*n^2-4*n+4)*x^2+(4*n^2-4*n-2)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 - (2*n^3 - 9*n^2 - 5*n + 30)*x - (6*n^3 - 19*n^2 + 12*n + 4)*x^2)*P[n - 1, x] - (6*x - (4*n^2 - 4*n + 4)*x^2 + (4*n^2 - 4*n - 2)*x^3)*D[P[n - 1, x], x])/(-3*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(3*n - 6 - (2*n^3 - 9*n^2 - 5*n + 30)*x - (6*n^3 - 19*n^2 + 12*n + 4)*x^2)*v[i - 1] - (6*x - (4*n^2 - 4*n + 4)*x^2 + (4*n^2 - 4*n - 2)*x^3)*deriv(v[i - 1], x))/(-3*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-731bf5f7395212d6
Coefficients of one sign. Every nonzero row, n >= 1: 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-731bf5f7395212d6
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-731bf5f7395212d6
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-731bf5f7395212d6

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-731bf5f7395212d6

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-1375f2f1a11f9809 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-a5e3cd5a8b6d6a1a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-1375f2f1a11f9809 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023