A111125 → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-n - n x + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x - x^{2} + 4\,n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	5	2													7
4	14	14	3												31
5	30	54	27	4											115
6	55	154	132	44	5										390
7	91	364	468	260	65	6									1254
8	140	756	1350	1100	450	90	7								3893
9	204	1428	3366	3740	2210	714	119	8							11789
10	285	2508	7524	10868	8645	3990	1064	152	9						35045
11	385	4158	15444	28028	28665	17640	6664	1512	189	10					102695
12	506	6578	29601	65780	83720	65688	32844	10488	2070	230	11				297516
13	650	10010	53625	143000	221000	214200	135660	57000	15750	2750	275	12			853932
14	819	14742	92664	291720	537030	627912	488376	258552	93555	22770	3564	324	13		2432041
15	1015	21112	153816	563992	1217710	1686060	1573656	1018248	462231	146740	31900	4524	377	14	6881395

Generated b-files: A197649 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-a4555bce014ca1b5 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-2bf8c8295f2d680d

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 ?

A197649: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:   1;
   3:   5,    2;
   4:  14,   14,    3;
   5:  30,   54,   27,     4;
   6:  55,  154,  132,    44,    5;
   7:  91,  364,  468,   260,   65,    6;
   8: 140,  756, 1350,  1100,  450,   90,    7;
   9: 204, 1428, 3366,  3740, 2210,  714,  119,   8;
  10: 285, 2508, 7524, 10868, 8645, 3990, 1064, 152, 9;

%F Formula

The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2-n+(n^2-n)*x)*P(n-1,x) + ((4*n-4)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + (n^2 - n)*x)*P[n - 1, x] + ((4*n - 4)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - n + (n^2 - n)*x)*v[i - 1] + ((4*n - 4)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-05948cabe35ede38
Coefficients of one sign. Every nonzero row, n >= 1: 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-05948cabe35ede38
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-05948cabe35ede38
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-05948cabe35ede38

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-05948cabe35ede38

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-0bb8ed02ac535b32 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-19ca7a5e6e7a3cfc 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