A053117 → x ↦ x+1 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	2														2
3	8	8													16
4	20	48	24												92
5	40	168	192	64											464
6	70	448	864	640	160										2182
7	112	1008	2880	3520	1920	384									9824
8	168	2016	7920	14080	12480	5376	896								42936
9	240	3696	19008	45760	58240	40320	14336	2048							183648
10	330	6336	41184	128128	218400	215040	121856	36864	4608						772746
11	440	10296	82368	320320	698880	913920	731136	350208	92160	10240					3209968
12	572	16016	154440	732160	1980160	3290112	3472896	2334720	967680	225280	22528				13196564
13	728	24024	274560	1555840	5091840	10418688	13891584	12257280	7096320	2590720	540672	49152			53791408
14	910	34944	466752	3111680	12093120	29767680	48620544	53932032	40803840	20725760	6758400	1277952	106496		217700110
15	1120	49504	763776	5912192	26873600	78140160	152807424	206739456	195858432	129536000	58572800	17252352	2981888	229376	875718080

Technical details

Node id: n-38e29c80ca926386 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5a38d53f8ebfe52d

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:   2;
   3:   8,    8;
   4:  20,   48,    24;
   5:  40,  168,   192,     64;
   6:  70,  448,   864,    640,    160;
   7: 112, 1008,  2880,   3520,   1920,    384;
   8: 168, 2016,  7920,  14080,  12480,   5376,    896;
   9: 240, 3696, 19008,  45760,  58240,  40320,  14336,  2048;
  10: 330, 6336, 41184, 128128, 218400, 215040, 121856, 36864, 4608;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((n + 1 + (n + 1)*x)*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], x))/(n - 2)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing or…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

chebyshev_u → x ↦ x+1 → D

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