A110441 → 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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	7	12	3						22
3	15	46	27	4					92
4	31	144	144	48	5				372
5	63	402	594	328	75	6			1468
6	127	1044	2097	1680	625	108	7		5688
7	255	2582	6669	7180	3825	1062	147	8	21728

Show all 15 stored rows

Generated b-files: A053297 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-d391e9db43dc4133 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9b3d5c5394a7d2f3

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

OEIS matches ?

A053297: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:    3,     2;
  2:    7,    12,     3;
  3:   15,    46,    27,     4;
  4:   31,   144,   144,    48,     5;
  5:   63,   402,   594,   328,    75,     6;
  6:  127,  1044,  2097,  1680,   625,   108,     7;
  7:  255,  2582,  6669,  7180,  3825,  1062,   147,    8;
  8:  511,  6168, 19686, 27072, 19200,  7560,  1666,  192,   9;
  9: 1023, 14362, 54972, 93104, 83790, 43596, 13524, 2464, 243, 10;

%F Formula

The row polynomials satisfy -(4*n+6*n*x-2*n^3*x^2)*P(n,x) = -(6*n+6+(n^2+13*n+12)*x-(3*n^3+3*n^2-6*n-6)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (1+9*x-(n^2+n-19)*x^2-(6*n^2+6*n-3)*x^3-(n^2+n)*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[(-(6*n + 6 + (n^2 + 13*n + 12)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (1 + 9*x - (n^2 + n - 19)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-4*n - 6*n*x + 2*n^3*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] = (-(6*n + 6 + (n^2 + 13*n + 12)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (1 + 9*x - (n^2 + n - 19)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-4*n - 6*n*x + 2*n^3*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-ec8938d063ea19e6
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-ec8938d063ea19e6
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-ec8938d063ea19e6
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-ec8938d063ea19e6

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-ec8938d063ea19e6

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangular array formed by the Mersenne numbers.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-c01294f1353c0463 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-02d1e2cf79c791bd root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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