A114687 → 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

\(\left(4 - 10\,x + 4\,x^{2} - 2\,n + 3\,n x + 2\,n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(4 - 10\,x + 4\,x^{2} + 6\,n + n x - 16\,n x^{2} + 12\,n x^{3} + 6\,n^{2} + 24\,n^{2} x^{2} + 2\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 6\,x^{2} + 8\,x^{3} - 8\,x^{4} + 2\,n x + 4\,n x^{2} - 16\,n x^{3} + 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	7	14	4						25
3	13	60	48	8					129
4	21	180	320	144	16				681
5	31	430	1400	1360	400	32			3653
6	43	882	4620	8120	5040	1056	64		19825
7	57	1624	12544	35280	39200	17024	2688	128	108545

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-c5d3384acf72de77 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-770cb16f18256565

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 ?

A002002: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  7,   14,     4;
  3: 13,   60,    48,      8;
  4: 21,  180,   320,    144,     16;
  5: 31,  430,  1400,   1360,    400,      32;
  6: 43,  882,  4620,   8120,   5040,    1056,     64;
  7: 57, 1624, 12544,  35280,  39200,   17024,   2688,    128;
  8: 73, 2760, 29568, 122304, 216384,  166656,  53760,   6656,   256;
  9: 91, 4410, 62640, 358848, 931392, 1137024, 645120, 161280, 16128, 512;

%F Formula

The row polynomials satisfy (2*n^3+2*n^2-2*n+4+(4*n^2+3*n-10)*x+(2*n+4)*x^2)*P(n,x) = (2*n^3+6*n^2+6*n+4+(12*n^3+n-10)*x+(24*n^2-16*n+4)*x^2+12*n*x^3)*P(n-1,x) + ((4*n^2+2*n+2)*x-(8*n^2-4*n+6)*x^2-(16*n-8)*x^3-8*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 + 6*n^2 + 6*n + 4 + (12*n^3 + n - 10)*x + (24*n^2 - 16*n + 4)*x^2 + 12*n*x^3)*P[n - 1, x] + ((4*n^2 + 2*n + 2)*x - (8*n^2 - 4*n + 6)*x^2 - (16*n - 8)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n^2 - 2*n + 4 + (4*n^2 + 3*n - 10)*x + (2*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] = ((2*n^3 + 6*n^2 + 6*n + 4 + (12*n^3 + n - 10)*x + (24*n^2 - 16*n + 4)*x^2 + 12*n*x^3)*v[i - 1] + ((4*n^2 + 2*n + 2)*x - (8*n^2 - 4*n + 6)*x^2 - (16*n - 8)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n^2 - 2*n + 4 + (4*n^2 + 3*n - 10)*x + (2*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-9941cb9a817a797c
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-9941cb9a817a797c
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-9941cb9a817a797c
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-9941cb9a817a797c

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-9941cb9a817a797c

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-0f37534727a0e0cd 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