A108838 → 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

\(\left(9 - 9\,x + 3\,x^{2} + 9\,n - 6\,n x + 4\,n x^{2} + 2\,n^{2} + 5\,n^{2} x + n^{2} x^{2} + 3\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(18 - 18\,x + 6\,x^{2} + 21\,n + 3\,n x - 5\,n x^{2} + 5\,n x^{3} + 14\,n^{2} + 10\,n^{2} x + 5\,n^{2} x^{2} + 3\,n^{2} x^{3} + 6\,n^{3} + 6\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(6\,x - 9\,x^{2} + 6\,x^{3} - 3\,x^{4} + 6\,n x - 2\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} + 4\,n^{2} x - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	5	2							7
2	12	12	2						26
3	25	50	21	2					98
4	46	160	132	32	2				372
5	77	420	595	280	45	2			1419
6	120	952	2100	1680	520	60	2		5434
7	177	1932	6174	7644	3990	882	77	2	20878

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-88849492685a3f59 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9f01c67d4f0b38b0

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 ?

A126223: 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:   2;
  1:   5,    2;
  2:  12,   12,     2;
  3:  25,   50,    21,     2;
  4:  46,  160,   132,    32,      2;
  5:  77,  420,   595,   280,     45,     2;
  6: 120,  952,  2100,  1680,    520,    60,     2;
  7: 177, 1932,  6174,  7644,   3990,   882,    77,    2;
  8: 250, 3600, 15792, 28224,  22680,  8400,  1400,   96,   2;
  9: 341, 6270, 36234, 88704, 102564, 58212, 16170, 2112, 117, 2;

%F Formula

The row polynomials satisfy (n^4+3*n^3+2*n^2+9*n+9+(2*n^3+5*n^2-6*n-9)*x+(n^2+4*n+3)*x^2)*P(n,x) = (n^4+6*n^3+14*n^2+21*n+18+(3*n^4+6*n^3+10*n^2+3*n-18)*x+(6*n^3+5*n^2-5*n+6)*x^2+(3*n^2+5*n)*x^3)*P(n-1,x) + ((2*n^3+4*n^2+6*n+6)*x-(2*n^3+2*n+9)*x^2-(4*n^2+2*n-6)*x^3-(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 6*n^3 + 14*n^2 + 21*n + 18 + (3*n^4 + 6*n^3 + 10*n^2 + 3*n - 18)*x + (6*n^3 + 5*n^2 - 5*n + 6)*x^2 + (3*n^2 + 5*n)*x^3)*P[n - 1, x] + ((2*n^3 + 4*n^2 + 6*n + 6)*x - (2*n^3 + 2*n + 9)*x^2 - (4*n^2 + 2*n - 6)*x^3 - (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^4 + 3*n^3 + 2*n^2 + 9*n + 9 + (2*n^3 + 5*n^2 - 6*n - 9)*x + (n^2 + 4*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 6*n^3 + 14*n^2 + 21*n + 18 + (3*n^4 + 6*n^3 + 10*n^2 + 3*n - 18)*x + (6*n^3 + 5*n^2 - 5*n + 6)*x^2 + (3*n^2 + 5*n)*x^3)*v[i - 1] + ((2*n^3 + 4*n^2 + 6*n + 6)*x - (2*n^3 + 2*n + 9)*x^2 - (4*n^2 + 2*n - 6)*x^3 - (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^4 + 3*n^3 + 2*n^2 + 9*n + 9 + (2*n^3 + 5*n^2 - 6*n - 9)*x + (n^2 + 4*n + 3)*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-2abc7f1e7ef64bbe
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-2abc7f1e7ef64bbe
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-2abc7f1e7ef64bbe
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-2abc7f1e7ef64bbe

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-2abc7f1e7ef64bbe

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a4549745e8cd4154 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