A257625 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(81 - 432\,x + 1053\,x^{2} - 990\,x^{3} + 216\,x^{4} - 54\,n + 324\,n x - 1098\,n x^{2} + 936\,n x^{3} - 144\,n x^{4} - 72\,n^{2} x + 468\,n^{2} x^{2} - 216\,n^{2} x^{3} - 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(54\,x - 396\,x^{2} + 810\,x^{3} - 612\,x^{4} + 144\,x^{5} - 36\,n x + 324\,n x^{2} - 504\,n x^{3} + 216\,n x^{4} - 72\,n^{2} x^{2} + 72\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-54 + 168\,x - 138\,x^{2} + 24\,x^{3} + 24\,n - 108\,n x + 60\,n x^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 54\,x^{2} - 24\,x^{3} + 12\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	3							9
2	63	72	9						144
3	648	1863	702	27					3240
4	6237	43416	37098	6480	81				93312
5	57834	875043	1634904	639090	58806	243			3265920
6	527067	15898032	59185323	47909880	10317537	530712	729		134369280
7	4767660	270079191	1836699462	2840071005	1231298496	161249697	4780782	2187	6348948480

All 15 rows

Technical details

Node id: n-c63b8c85c56b948d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-04cae45d42f04794

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:         6,           3;
  2:        63,          72,             9;
  3:       648,        1863,           702,            27;
  4:      6237,       43416,         37098,          6480,            81;
  5:     57834,      875043,       1634904,        639090,         58806,           243;
  6:    527067,    15898032,      59185323,      47909880,      10317537,        530712,          729;
  7:   4767660,   270079191,    1836699462,    2840071005,    1231298496,     161249697,      4780782,        2187;
  8:  42994233,  4394400336,   51093630036,  138997041984,  112202639694,   29360212560,   2476620036,    43040160,      6561;
  9: 387243342, 69527304099, 1316404078848, 5892325476084, 8197571194812, 3929929943058, 669268294416, 37667829492, 387400806, 19683;

%F Formula

The row polynomials satisfy (2*n-5+4*x)*P(n,x) = -(54*n-81+(72*n^2-324*n+432)*x+(72*n^3-468*n^2+1098*n-1053)*x^2+(216*n^2-936*n+990)*x^3+(144*n-216)*x^4)*P(n-2,x) - ((36*n-54)*x+(72*n^2-324*n+396)*x^2-(72*n^2-504*n+810)*x^3-(216*n-612)*x^4-144*x^5)*d/dx P(n-2,x) + (24*n-54+(24*n^2-108*n+168)*x+(60*n-138)*x^2+24*x^3)*P(n-1,x) + ((12*n-30)*x-(12*n-54)*x^2-24*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n - 81 + (72*n^2 - 324*n + 432)*x + (72*n^3 - 468*n^2 + 1098*n - 1053)*x^2 + (216*n^2 - 936*n + 990)*x^3 + (144*n - 216)*x^4)*P[n - 2, x] - ((36*n - 54)*x + (72*n^2 - 324*n + 396)*x^2 - (72*n^2 - 504*n + 810)*x^3 - (216*n - 612)*x^4 - 144*x^5)*D[P[n - 2, x], x] + (24*n - 54 + (24*n^2 - 108*n + 168)*x + (60*n - 138)*x^2 + 24*x^3)*P[n - 1, x] + ((12*n - 30)*x - (12*n - 54)*x^2 - 24*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 4*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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(54*n - 81 + (72*n^2 - 324*n + 432)*x + (72*n^3 - 468*n^2 + 1098*n - 1053)*x^2 + (216*n^2 - 936*n + 990)*x^3 + (144*n - 216)*x^4)*v[i - 2] - ((36*n - 54)*x + (72*n^2 - 324*n + 396)*x^2 - (72*n^2 - 504*n + 810)*x^3 - (216*n - 612)*x^4 - 144*x^5)*deriv(v[i - 2], x) + (24*n - 54 + (24*n^2 - 108*n + 168)*x + (60*n - 138)*x^2 + 24*x^3)*v[i - 1] + ((12*n - 30)*x - (12*n - 54)*x^2 - 24*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 4*x)); 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 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.