A082985 → 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

\(\left(9\,x - 4\,x^{2} + 4\,n - 26\,n x + 8\,n x^{2} - 12\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(4 + 20\,x + 11\,x^{2} - 12\,x^{3} - 8\,n - 44\,n x - 46\,n x^{2} + 24\,n x^{3} - 4\,n^{2} - 4\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 9\,x^{2} - 16\,x^{4} - 2\,n x - 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	6	15	5						26
3	8	35	35	7					85
4	10	63	117	66	9				265
5	12	99	275	297	110	11			804
6	14	143	533	884	637	169	13		2393
7	16	195	915	2075	2340	1218	245	15	7019

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-a8f047ca6c0b04d7 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d672f84bcb723328

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,   3;
  2:  6,  15,    5;
  3:  8,  35,   35,    7;
  4: 10,  63,  117,   66,     9;
  5: 12,  99,  275,  297,   110,    11;
  6: 14, 143,  533,  884,   637,   169,    13;
  7: 16, 195,  915, 2075,  2340,  1218,   245,   15;
  8: 18, 255, 1445, 4182,  6545,  5406,  2142,  340,  17;
  9: 20, 323, 2147, 7581, 15314, 17765, 11286, 3534, 456, 19;

%F Formula

The row polynomials satisfy (8*n^3-12*n^2+4*n+(16*n^2-26*n+9)*x+(8*n-4)*x^2)*P(n,x) = (8*n^3-4*n^2-8*n+4+(24*n^3-4*n^2-44*n+20)*x+(48*n^2-46*n+11)*x^2+(24*n-12)*x^3)*P(n-1,x) - ((4*n^2+2*n-2)*x+(16*n^2+16*n-9)*x^2+32*n*x^3+16*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[((8*n^3 - 4*n^2 - 8*n + 4 + (24*n^3 - 4*n^2 - 44*n + 20)*x + (48*n^2 - 46*n + 11)*x^2 + (24*n - 12)*x^3)*P[n - 1, x] - ((4*n^2 + 2*n - 2)*x + (16*n^2 + 16*n - 9)*x^2 + 32*n*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 12*n^2 + 4*n + (16*n^2 - 26*n + 9)*x + (8*n - 4)*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] = ((8*n^3 - 4*n^2 - 8*n + 4 + (24*n^3 - 4*n^2 - 44*n + 20)*x + (48*n^2 - 46*n + 11)*x^2 + (24*n - 12)*x^3)*v[i - 1] - ((4*n^2 + 2*n - 2)*x + (16*n^2 + 16*n - 9)*x^2 + 32*n*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 12*n^2 + 4*n + (16*n^2 - 26*n + 9)*x + (8*n - 4)*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-96ae67f48970d85a
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-96ae67f48970d85a
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-96ae67f48970d85a
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-96ae67f48970d85a

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-96ae67f48970d85a

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4fa62233b528a616 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