A110441 → D+1

nonnegative coefficients

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

Recurrence

\(\left(14\,n - 6\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(23 - 7\,x - 7\,x^{2} - x^{3} + 29\,n - 5\,n x - 7\,n x^{2} - n x^{3} + 9\,n^{2} + 3\,n^{2} x + 3\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(2 + 7\,x - 29\,x^{2} - 11\,x^{3} - x^{4} + n + 6\,n x + n x^{2} + n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	1							5
2	13	8	1						22
3	38	41	12	1					92
4	103	168	84	16	1				372
5	264	597	444	142	20	1			1468
6	649	1920	1959	920	215	24	1		5688
7	1546	5737	7608	4855	1650	303	28	1	21728

Show all 15 stored rows

Generated b-files: A053297 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-e70bc71f808d87be · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4f3de3e72d370144

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:    4,     1;
  2:   13,     8,     1;
  3:   38,    41,    12,     1;
  4:  103,   168,    84,    16,     1;
  5:  264,   597,   444,   142,    20,     1;
  6:  649,  1920,  1959,   920,   215,    24,    1;
  7: 1546,  5737,  7608,  4855,  1650,   303,   28,   1;
  8: 3595, 16208, 26866, 22128, 10140,  2688,  406,  32,  1;
  9: 8204, 43829, 88152, 90308, 53088, 18858, 4088, 524, 36, 1;

%F Formula

The row polynomials satisfy (2*n^3+14*n-6*n*x-2*n*x^2)*P(n,x) = (3*n^3+9*n^2+29*n+23+(n^3+3*n^2-5*n-7)*x-(7*n+7)*x^2-(n+1)*x^3)*P(n-1,x) + (n^2+n+2+(6*n^2+6*n+7)*x+(n^2+n-29)*x^2-11*x^3-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^3 + 9*n^2 + 29*n + 23 + (n^3 + 3*n^2 - 5*n - 7)*x - (7*n + 7)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (n^2 + n + 2 + (6*n^2 + 6*n + 7)*x + (n^2 + n - 29)*x^2 - 11*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 14*n - 6*n*x - 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] = ((3*n^3 + 9*n^2 + 29*n + 23 + (n^3 + 3*n^2 - 5*n - 7)*x - (7*n + 7)*x^2 - (n + 1)*x^3)*v[i - 1] + (n^2 + n + 2 + (6*n^2 + 6*n + 7)*x + (n^2 + n - 29)*x^2 - 11*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 14*n - 6*n*x - 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-9b8106d62ea669b5
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-9b8106d62ea669b5
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-9b8106d62ea669b5
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-9b8106d62ea669b5

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-9b8106d62ea669b5

Laguerre’s theorem on differential operators [SP14] 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

[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