A229032 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-7 - 4\,x + 9\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-28 - 30\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	11	4							15
3	36	16							52
4	149	104	16						269
5	550	464	96						1110
6	2143	2252	720	64					5179
7	8136	9952	3968	512					22568
8	31273	44240	21984	4352	256				102105
9	119498	190688	111552	28160	2560				452458
10	457907	816148	553952	174720	24320	1024			2028071
11	1752300	3445104	2651520	992768	179200	12288			9033180
12	6709949	14434872	12445296	5424384	1227520	129024	4096		40375141
13	25685998	59994672	57231648	28404224	7680512	1060864	57344		180115262
14	98341639	247873244	259319856	144560320	45651200	7934976	659456	16384	804357075

Technical details

Node id: n-e75fe66ef251ba0e · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-cb987b668813c912

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      2;
  2:     11,      4;
  3:     36,     16;
  4:    149,    104,     16;
  5:    550,    464,     96;
  6:   2143,   2252,    720,    64;
  7:   8136,   9952,   3968,   512;
  8:  31273,  44240,  21984,  4352,  256;
  9: 119498, 190688, 111552, 28160, 2560;

%F Formula

The row polynomials satisfy n*P(n,x) = (9*n-7+(4*n-4)*x)*P(n-1,x) - (28+30*x+8*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 7 + (4*n - 4)*x)*P[n - 1, x] - (28 + 30*x + 8*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((9*n - 7 + (4*n - 4)*x)*v[i - 1] - (28 + 30*x + 8*x^2)*deriv(v[i - 1], x))/(n)); 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 shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).?
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).?
    
    ↑ Gamma criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).?
    
    ↑ Gamma criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023