A123519 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(9 + 12\,x + 9\,n + 10\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,x + 4\,x^{2} + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	6	8								14
1	12	40	24							76
2	20	120	168	64						372
3	30	280	672	576	160					1718
4	42	560	2016	2880	1760	384				7642
5	56	1008	5040	10560	10560	4992	896			33112
6	72	1680	11088	31680	45760	34944	13440	2048		140712
7	90	2640	22176	82368	160160	174720	107520	34816	4608	589098

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-782f87aa4c7db77e · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-c015fe9c958530bb

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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:   6,    8;
  1:  12,   40,    24;
  2:  20,  120,   168,     64;
  3:  30,  280,   672,    576,     160;
  4:  42,  560,  2016,   2880,    1760,     384;
  5:  56, 1008,  5040,  10560,   10560,    4992,     896;
  6:  72, 1680, 11088,  31680,   45760,   34944,   13440,    2048;
  7:  90, 2640, 22176,  82368,  160160,  174720,  107520,   34816,   4608;
  8: 110, 3960, 41184, 192192,  480480,  698880,  609280,  313344,  87552,  10240;
  9: 132, 5720, 72072, 411840, 1281280, 2376192, 2741760, 1984512, 875520, 215040, 22528;

%F Formula

The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (2*n^2+9*n+9+(2*n^2+10*n+12)*x)*P(n-1,x) + ((4*n+8)*x+(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 8*x.

%t Mathematica

P[0, x_] := 6 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 9*n + 9 + (2*n^2 + 10*n + 12)*x)*P[n - 1, x] + ((4*n + 8)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 9*n + 9 + (2*n^2 + 10*n + 12)*x)*v[i - 1] + ((4*n + 8)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*n + 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-2a3f07ca8f61464b
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-2a3f07ca8f61464b
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-2a3f07ca8f61464b
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-2a3f07ca8f61464b

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-2a3f07ca8f61464b

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef 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