basis-map coefficients of A371080, shifted → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 30\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 4\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	14	18	4						36
3	106	174	72	8					360
4	1116	2108	1164	240	16				4644
5	15068	31308	20620	5800	720	32			73548
6	248368	553920	412416	141160	24240	2016	64		1382184
7	4835472	11401472	9307536	3660552	775040	90048	5376	128	30075624

All 15 rows

Technical details

Node id: n-5d19d732b0c7e06c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8032bdd3ad4ca3e6

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:          1;
  1:          3,          2;
  2:         14,         18,          4;
  3:        106,        174,         72,          8;
  4:       1116,       2108,       1164,        240,        16;
  5:      15068,      31308,      20620,       5800,       720,        32;
  6:     248368,     553920,     412416,     141160,     24240,      2016,       64;
  7:    4835472,   11401472,    9307536,    3660552,    775040,     90048,     5376,    128;
  8:  108593504,  267961152,  235241296,  102794720,  25290384,   3657472,   307328,  13824,   256;
  9: 2763526240, 7084714176, 6600666672, 3135940800, 865348848, 146633760, 15447936, 983808, 34560, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(12*n-20)*x+4*x^2)*P(n-2,x) + (6*n-7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*P[n - 2, x] + (6*n - 7 + 4*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] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*v[i - 2] + (6*n - 7 + 4*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) 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
  - Geometric basis map (elementary) 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 (above)

References

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