Hadamard-factor parent of A155495 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(n + 4\,n x + 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1 + 8\,x + 8\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	1	8	8								17
2	1	18	48	32							99
3	1	32	160	256	128						577
4	1	50	400	1120	1280	512					3363
5	1	72	840	3584	6912	6144	2048				19601
6	1	98	1568	9408	26880	39424	28672	8192			114243
7	1	128	2688	21504	84480	180224	212992	131072	32768		665857
8	1	162	4320	44352	228096	658944	1118208	1105920	589824	131072	3880899

All 15 rows

Technical details

Node id: n-537424a659b74b6c · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-607690c148a1ebbd

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,   8,    8;
   2: 1,  18,   48,     32;
   3: 1,  32,  160,    256,     128;
   4: 1,  50,  400,   1120,    1280,     512;
   5: 1,  72,  840,   3584,    6912,    6144,     2048;
   6: 1,  98, 1568,   9408,   26880,   39424,    28672,     8192;
   7: 1, 128, 2688,  21504,   84480,  180224,   212992,   131072,    32768;
   8: 1, 162, 4320,  44352,  228096,  658944,  1118208,  1105920,   589824,   131072;
   9: 1, 200, 6600,  84480,  549120, 2050048,  4659200,  6553600,  5570560,  2621440,   524288;
  10: 1, 242, 9680, 151008, 1208064, 5637632, 16400384, 30638080, 36765696, 27394048, 11534336, 2097152;

%F Formula

The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+4*n*x+5*n*x^2+2*n*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 8*x^2.

%t Mathematica

P[1, x_] := 1 + 8*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4*n*x + 5*n*x^2 + 2*n*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = ((n + 4*n*x + 5*n*x^2 + 2*n*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
  - 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

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