A372873 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(1 - 7\,x - 23\,x^{2} - 21\,x^{3} - 6\,x^{4} - n + 7\,n x + 23\,n x^{2} + 21\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x - 12\,x^{3} - 16\,x^{4} - 6\,x^{5}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	1	5													6
3	0	0	6	14												20
4	0	0	1	26	41											68
5	0	0	0	10	100	122										232
6	0	0	0	1	63	363	365									792
7	0	0	0	0	14	322	1274	1094								2704
8	0	0	0	0	1	116	1462	4372	3281							9232
9	0	0	0	0	0	18	744	6156	14760	9842						31520
10	0	0	0	0	0	1	185	4090	24610	49205	29525					107616
11	0	0	0	0	0	0	22	1430	20284	94732	162382	88574				367424
12	0	0	0	0	0	0	1	270	9255	93476	354303	531438	265721			1254464
13	0	0	0	0	0	0	0	26	2444	53014	407784	1295398	1727180	797162		4283008
14	0	0	0	0	0	0	0	1	371	18221	277767	1705011	4650121	5580127	2391485	14623104

Technical details

Node id: n-7556acd10c14a2be · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-97ff5e3c0ee49bcd

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: 0, 2;
  2: 0, 1, 5;
  3: 0, 0, 6, 14;
  4: 0, 0, 1, 26,  41;
  5: 0, 0, 0, 10, 100, 122;
  6: 0, 0, 0,  1,  63, 363,  365;
  7: 0, 0, 0,  0,  14, 322, 1274, 1094;
  8: 0, 0, 0,  0,   1, 116, 1462, 4372,  3281;
  9: 0, 0, 0,  0,   0,  18,  744, 6156, 14760, 9842;

%F Formula

The row polynomials satisfy (2*n-2+(4*n-4)*x+(2*n-2)*x^2)*P(n,x) = -(n-1-(7*n-7)*x-(23*n-23)*x^2-(21*n-21)*x^3-(6*n-6)*x^4)*P(n-1,x) + (2*x-12*x^3-16*x^4-6*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 - (7*n - 7)*x - (23*n - 23)*x^2 - (21*n - 21)*x^3 - (6*n - 6)*x^4)*P[n - 1, x] + (2*x - 12*x^3 - 16*x^4 - 6*x^5)*D[P[n - 1, x], x])/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*x^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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 - (7*n - 7)*x - (23*n - 23)*x^2 - (21*n - 21)*x^3 - (6*n - 6)*x^4)*v[i - 1] + (2*x - 12*x^3 - 16*x^4 - 6*x^5)*deriv(v[i - 1], x))/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*x^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, 0]
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 Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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