A142075 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	3							4
3	1	10	13						24
4	1	25	91	75					192
5	1	56	426	896	541				1920
6	1	119	1674	6734	9829	4683			23040
7	1	246	5979	40804	108039	120198	47293		322560
8	1	501	20157	218257	931347	1815951	1628911	545835	5160960

All 15 rows

Technical details

Node id: n-d38ac1988a9f3660 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-31922ed48acb43c6

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 -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;
   2: 1,    3;
   3: 1,   10,     13;
   4: 1,   25,     91,      75;
   5: 1,   56,    426,     896,      541;
   6: 1,  119,   1674,    6734,     9829,      4683;
   7: 1,  246,   5979,   40804,   108039,    120198,     47293;
   8: 1,  501,  20157,  218257,   931347,   1815951,   1628911,    545835;
   9: 1, 1012,  65488, 1077604,  6940090,  21082204,  32355928,  24287692,   7087261;
  10: 1, 2035, 207604, 5037868, 46947022, 208222234, 486156100, 613463356, 395661817, 102247563;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + (n + 3)*x + (3*n + 1)*x^2 + (3*n - 3)*x^3 + (n - 2)*x^4)*v[i - 1] + (x + 2*x^2 - 2*x^4 - x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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 Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A156365 → 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