basis-map coefficients of A371080, shifted → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 30\,x - 9\,x^{2} + 30\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	19	33	9						61
3	145	333	189	27					694
4	1534	4116	3213	918	81				9862
5	20782	61872	58410	23355	4050	243			168712
6	343432	1103724	1187271	584955	143370	16767	729		3380248
7	6699928	22856628	27097119	15453963	4723110	785862	66339	2187	77685136

All 15 rows

Technical details

Node id: n-7794429e5748ba6a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0a9a0c55c6b499ab

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:          4,           3;
  2:         19,          33,           9;
  3:        145,         333,         189,          27;
  4:       1534,        4116,        3213,         918,         81;
  5:      20782,       61872,       58410,       23355,       4050,        243;
  6:     343432,     1103724,     1187271,      584955,     143370,      16767,       729;
  7:    6699928,    22856628,    27097119,    15453963,    4723110,     785862,     66339,     2187;
  8:  150718096,   539725704,   690601392,   439744032,  157244409,   32897340,   3970134,   253692,   6561;
  9: 3840985360, 14324418216, 19504239804, 13549330404, 5459168988, 1346748039, 205630488, 18856314, 944784, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 6*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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 6*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_A013610 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.