A232535 → 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

\(\left(9 - 27\,x + 6\,x^{2} - 24\,n + 48\,n x - 15\,n x^{2} + 2\,n x^{3} + 22\,n^{2} - 19\,n^{2} x + 4\,n^{2} x^{2} - 8\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(3 - 49\,x - 61\,x^{2} + 25\,x^{3} - 2\,x^{4} - 4\,n + 44\,n x + 99\,n x^{2} - 41\,n x^{3} + 6\,n x^{4} + 6\,n^{2} + 37\,n^{2} x - 51\,n^{2} x^{2} + 12\,n^{2} x^{3} - 8\,n^{3} - 22\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(12\,x + 22\,x^{2} - 56\,x^{3} + 26\,x^{4} - 4\,x^{5} + 8\,n x - 40\,n x^{2} + 40\,n x^{3} - 8\,n x^{4} - 16\,n^{2} x + 20\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	9	14	3						26
3	19	68	37	4					128
4	33	228	266	76	5				608
5	51	590	1278	756	135	6			2816
6	73	1282	4565	4884	1771	218	7		12800
7	99	2464	13195	22880	14729	3640	329	8	57344

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6eec6efe3fe38b73 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b61617a698b0fbfc

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:   3,    2;
  2:   9,   14,     3;
  3:  19,   68,    37,      4;
  4:  33,  228,   266,     76,      5;
  5:  51,  590,  1278,    756,    135,      6;
  6:  73, 1282,  4565,   4884,   1771,    218,     7;
  7:  99, 2464, 13195,  22880,  14729,   3640,   329,     8;
  8: 129, 4328, 32676,  84552,  87230,  37752,  6804,   472,   9;
  9: 163, 7098, 72012, 261528, 400010, 274924, 85884, 11832, 651, 10;

%F Formula

The row polynomials satisfy -(8*n^3-22*n^2+24*n-9-(2*n^3-19*n^2+48*n-27)*x-(4*n^2-15*n+6)*x^2-2*n*x^3)*P(n,x) = -(8*n^3-6*n^2+4*n-3+(22*n^3-37*n^2-44*n+49)*x-(6*n^3-51*n^2+99*n-61)*x^2-(12*n^2-41*n+25)*x^3-(6*n-2)*x^4)*P(n-1,x) - ((16*n^2-8*n-12)*x-(20*n^2-40*n+22)*x^2+(4*n^2-40*n+56)*x^3+(8*n-26)*x^4+4*x^5)*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 - 6*n^2 + 4*n - 3 + (22*n^3 - 37*n^2 - 44*n + 49)*x - (6*n^3 - 51*n^2 + 99*n - 61)*x^2 - (12*n^2 - 41*n + 25)*x^3 - (6*n - 2)*x^4)*P[n - 1, x] - ((16*n^2 - 8*n - 12)*x - (20*n^2 - 40*n + 22)*x^2 + (4*n^2 - 40*n + 56)*x^3 + (8*n - 26)*x^4 + 4*x^5)*D[P[n - 1, x], x])/(-8*n^3 + 22*n^2 - 24*n + 9 + (2*n^3 - 19*n^2 + 48*n - 27)*x + (4*n^2 - 15*n + 6)*x^2 + 2*n*x^3)]];
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 - 6*n^2 + 4*n - 3 + (22*n^3 - 37*n^2 - 44*n + 49)*x - (6*n^3 - 51*n^2 + 99*n - 61)*x^2 - (12*n^2 - 41*n + 25)*x^3 - (6*n - 2)*x^4)*v[i - 1] - ((16*n^2 - 8*n - 12)*x - (20*n^2 - 40*n + 22)*x^2 + (4*n^2 - 40*n + 56)*x^3 + (8*n - 26)*x^4 + 4*x^5)*deriv(v[i - 1], x))/(-8*n^3 + 22*n^2 - 24*n + 9 + (2*n^3 - 19*n^2 + 48*n - 27)*x + (4*n^2 - 15*n + 6)*x^2 + 2*n*x^3)); 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-e2dffcc245443c80
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-e2dffcc245443c80
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-e2dffcc245443c80
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-e2dffcc245443c80

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-e2dffcc245443c80

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) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/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-a75bd3d9d36c6f35 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