A249252 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(2 + 8\,x + 2\,x^{2} - 6\,n - 12\,n x - 2\,n x^{2} + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(38 + 76\,x + 42\,x^{2} + 4\,x^{3} - 17\,n - 31\,n x - 15\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(17 + 48\,x + 46\,x^{2} + 16\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 5 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	1							6
2	17	10	1						28
3	61	59	15	1					136
4	217	292	126	20	1				656
5	773	1333	818	218	25	1			3168
6	2753	5790	4623	1764	335	30	1		15296
7	9805	24335	23961	11987	3255	477	35	1	73856

All 15 rows

Technical details

Node id: n-4b2d537960f762e0 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-78b8e1ecee5b9454

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:      5,      1;
  2:     17,     10,      1;
  3:     61,     59,     15,      1;
  4:    217,    292,    126,     20,      1;
  5:    773,   1333,    818,    218,     25,     1;
  6:   2753,   5790,   4623,   1764,    335,    30,    1;
  7:   9805,  24335,  23961,  11987,   3255,   477,   35,   1;
  8:  34921,  99896, 117044,  72696,  25950,  5416,  644,  40,  1;
  9: 124373, 402889, 547620, 407028, 181030, 49662, 8372, 836, 45, 1;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = (4*n^2-6*n+2+(4*n^2-12*n+8)*x-(2*n-2)*x^2)*P(n-1,x) - (17*n-38+(31*n-76)*x+(15*n-42)*x^2+(n-4)*x^3)*d/dx P(n-1,x) + (17+48*x+46*x^2+16*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (4*n^2 - 12*n + 8)*x - (2*n - 2)*x^2)*P[n - 1, x] - (17*n - 38 + (31*n - 76)*x + (15*n - 42)*x^2 + (n - 4)*x^3)*D[P[n - 1, x], x] + (17 + 48*x + 46*x^2 + 16*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
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] = 5 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 6*n + 2 + (4*n^2 - 12*n + 8)*x - (2*n - 2)*x^2)*v[i - 1] - (17*n - 38 + (31*n - 76)*x + (15*n - 42)*x^2 + (n - 4)*x^3)*deriv(v[i - 1], x) + (17 + 48*x + 46*x^2 + 16*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(2*n - 2)); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A249252 → x ↦ x/(x+1)

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