A011973 → x ↦ x−1/4 → D+1

nonnegative coefficients

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

Recurrence

\(\left(7\,n - 22\,n x + 8\,n x^{2} - 6\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(3 - 26\,x + 64\,x^{2} - 32\,x^{3} + n + 14\,n x - 80\,n x^{2} + 32\,n x^{3} - 24\,n^{2} x + 32\,n^{2} x^{2} + 2\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 20\,x^{2} + 64\,x^{3} - 64\,x^{4} - 4\,n x + 32\,n x^{2} - 64\,n x^{3} + 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	7	4			11
3	20	16			36
4	45	72	16		133
5	86	272	96		454
6	147	812	528	64	1551
7	232	2016	2432	512	5192

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-92abebe007fb10d9 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-74a850e0b5b3645e

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   2;
  2:   7,    4;
  3:  20,   16;
  4:  45,   72,    16;
  5:  86,  272,    96;
  6: 147,  812,   528,    64;
  7: 232, 2016,  2432,   512;
  8: 345, 4368,  8928,  3328,  256;
  9: 490, 8544, 27072, 17920, 2560;

%F Formula

The row polynomials satisfy (2*n^3-6*n^2+7*n+(8*n^2-22*n)*x+8*n*x^2)*P(n,x) = (2*n^3+n+3+(8*n^3-24*n^2+14*n-26)*x+(32*n^2-80*n+64)*x^2+(32*n-32)*x^3)*P(n-1,x) + ((4*n^2-4*n+2)*x-(16*n^2-32*n+20)*x^2-(64*n-64)*x^3-64*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[((2*n^3 + n + 3 + (8*n^3 - 24*n^2 + 14*n - 26)*x + (32*n^2 - 80*n + 64)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 2)*x - (16*n^2 - 32*n + 20)*x^2 - (64*n - 64)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 7*n + (8*n^2 - 22*n)*x + 8*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] = ((2*n^3 + n + 3 + (8*n^3 - 24*n^2 + 14*n - 26)*x + (32*n^2 - 80*n + 64)*x^2 + (32*n - 32)*x^3)*v[i - 1] + ((4*n^2 - 4*n + 2)*x - (16*n^2 - 32*n + 20)*x^2 - (64*n - 64)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 7*n + (8*n^2 - 22*n)*x + 8*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-fe74f628d5055d01
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-fe74f628d5055d01
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-fe74f628d5055d01

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-fe74f628d5055d01

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - gamma_polynomial scale 1, shift 0 e-e2e7a7b682b47947
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    preserver_image reversal, shift 0 e-75f779345b39f910
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)

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