basis-map coefficients of A371080, shifted → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 40\,x - 16\,x^{2} + 30\,n - 24\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 8\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	4							4
2	20	32						52
3	156	384	192					732
4	1672	5088	4224	1024				12008
5	22856	79520	85440	35840	5120			228776
6	380256	1455168	1844160	1029120	261120	24576		4994400
7	7457824	30690240	43770048	29388800	10106880	1720320	114688	123248800

All 15 rows

Technical details

Node id: n-472235c6f6d3b99c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f8e7492a8b48ef64

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:          4;
  2:         20,          32;
  3:        156,         384,         192;
  4:       1672,        5088,        4224,        1024;
  5:      22856,       79520,       85440,       35840,        5120;
  6:     380256,     1455168,     1844160,     1029120,      261120,      24576;
  7:    7457824,    30690240,    43770048,    29388800,    10106880,    1720320,    114688;
  8:  168498368,   734847872,  1147082496,   879993856,   369295360,   86360064,  10551296,   524288;
  9: 4309836480, 19719183744, 33086149632, 28117619712, 13632353280, 3931619328, 666796032, 61341696, 2359296;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(24*n-40)*x+16*x^2)*P(n-2,x) + (6*n-7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*P[n - 2, x] + (6*n - 7 + 8*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (24*n - 40)*x + 16*x^2)*v[i - 2] + (6*n - 7 + 8*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_A013611 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.