basis-map coefficients of A371080, shifted → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 + 12\,x - x^{2} + 36\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	3	2													5
3	18	18	3												39
4	162	198	54	4											418
5	1944	2700	945	120	5										5714
6	29160	44388	18225	3060	225	6									95064
7	524880	857304	394632	79380	7875	378	7								1864456
8	11022480	19053144	9573228	2193156	264600	17388	588	8							42124592
9	264539520	479320416	258337188	65400048	9091845	734832	34398	864	9						1077459120
10	7142567040	13470730272	7694084700	2110250880	327229875	30750678	1786050	62640	1215	10					30777463360
11	214277011200	418407042240	251028636408	73566306000	12454709850	1315196190	89457291	3920400	106920	1650	11				971142388160
12	7071141369600	14235986416320	8911555564824	2762392946544	502963307550	58347126090	4486486158	231610104	7938810	173250	2178	12			33547112941440
13	254561089305600	526637793726720	342169979958144	111328220162016	21559670254980	2704052508300	229585148793	13465376496	546358527	15057900	268983	2808	13		1259204418129280
14	9927882482918400	21047996133953280	14134585908957696	4798027226262816	979987415146740	131329652129676	12108548729277	787532710536	36456531111	1194323130	27054027	402948	3549	14	51032742579123200

Technical details

Node id: n-be3fc0d186001aa8 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6e91ab2c0a184352

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 -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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         0;
  1:         1;
  2:         3,         2;
  3:        18,        18,         3;
  4:       162,       198,        54,        4;
  5:      1944,      2700,       945,      120,       5;
  6:     29160,     44388,     18225,     3060,     225,      6;
  7:    524880,    857304,    394632,    79380,    7875,    378,     7;
  8:  11022480,  19053144,   9573228,  2193156,  264600,  17388,   588,   8;
  9: 264539520, 479320416, 258337188, 65400048, 9091845, 734832, 34398, 864, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-36*n+36+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
  - Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A225470 → x ↦ x+1 → θ+1

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.