A119900 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

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	4													5
3	0	0	4	8												12
4	0	0	1	12	16											29
5	0	0	0	6	32	32										70
6	0	0	0	1	24	80	64									169
7	0	0	0	0	8	80	192	128								408
8	0	0	0	0	1	40	240	448	256							985
9	0	0	0	0	0	10	160	672	1024	512						2378
10	0	0	0	0	0	1	60	560	1792	2304	1024					5741
11	0	0	0	0	0	0	12	280	1792	4608	5120	2048				13860
12	0	0	0	0	0	0	1	84	1120	5376	11520	11264	4096			33461
13	0	0	0	0	0	0	0	14	448	4032	15360	28160	24576	8192		80782
14	0	0	0	0	0	0	0	1	112	2016	13440	42240	67584	53248	16384	195025

Technical details

Node id: n-5ad70a9d47b9fac2 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-6cf8c5bbd0a3960a

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, 4;
  3: 0, 0, 4,  8;
  4: 0, 0, 1, 12, 16;
  5: 0, 0, 0,  6, 32, 32;
  6: 0, 0, 0,  1, 24, 80,  64;
  7: 0, 0, 0,  0,  8, 80, 192, 128;
  8: 0, 0, 0,  0,  1, 40, 240, 448,  256;
  9: 0, 0, 0,  0,  0, 10, 160, 672, 1024, 512;

%F Formula

The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(n-1+(2*n-4)*x+(n-5)*x^2-2*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 >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (2*n - 4)*x + (n - 5)*x^2 - 2*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, 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] = (-(n - 1 + (2*n - 4)*x + (n - 5)*x^2 - 2*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 - 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 binary words of length n with k strictly increasing runs, for 0<=k<=n.?
  - 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