A125662 → 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

\(\left(6\,n - 6\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(7 - 11\,x - 7\,x^{2} - x^{3} + 13\,n - 9\,n x - 7\,n x^{2} - n x^{3} + 9\,n^{2} + 3\,n^{2} x + 3\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-10 - 37\,x - 37\,x^{2} - 11\,x^{3} - x^{4} + 5\,n + 6\,n x + n x^{2} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	1							5
2	14	8	1						23
3	46	43	12	1					102
4	145	192	87	16	1				441
5	444	768	492	146	20	1			1871
6	1331	2856	2391	1000	220	24	1		7823
7	3926	10085	10488	5725	1770	309	28	1	32332

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-9ab01f026cf977fc · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ab0d65133ac4f36a

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 ?

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:     4,      1;
  2:    14,      8,      1;
  3:    46,     43,     12,      1;
  4:   145,    192,     87,     16,     1;
  5:   444,    768,    492,    146,    20,     1;
  6:  1331,   2856,   2391,   1000,   220,    24,    1;
  7:  3926,  10085,  10488,   5725,  1770,   309,   28,   1;
  8: 11434,  34264,  42731,  28968, 11670,  2856,  413,  32,  1;
  9: 32960, 112991, 164676, 133918, 66948, 21315, 4312, 532, 36, 1;

%F Formula

The row polynomials satisfy (2*n^3+6*n-6*n*x-2*n*x^2)*P(n,x) = (3*n^3+9*n^2+13*n+7+(n^3+3*n^2-9*n-11)*x-(7*n+7)*x^2-(n+1)*x^3)*P(n-1,x) + (5*n^2+5*n-10+(6*n^2+6*n-37)*x+(n^2+n-37)*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 + 13*n + 7 + (n^3 + 3*n^2 - 9*n - 11)*x - (7*n + 7)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (5*n^2 + 5*n - 10 + (6*n^2 + 6*n - 37)*x + (n^2 + n - 37)*x^2 - 11*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*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 + 13*n + 7 + (n^3 + 3*n^2 - 9*n - 11)*x - (7*n + 7)*x^2 - (n + 1)*x^3)*v[i - 1] + (5*n^2 + 5*n - 10 + (6*n^2 + 6*n - 37)*x + (n^2 + n - 37)*x^2 - 11*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*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-c18e77b95eaee1ab
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-c18e77b95eaee1ab
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-c18e77b95eaee1ab
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-c18e77b95eaee1ab

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-c18e77b95eaee1ab

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci 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-af8f298ca8f41adf root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-c7df247a48c8552e 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