A107131 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(4 + 20\,x + 34\,x^{2} + 24\,x^{3} + 6\,x^{4} - 4\,n - 17\,n x - 28\,n x^{2} - 21\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 30\,x^{2} + 42\,x^{3} + 26\,x^{4} + 6\,x^{5}\right)\,P_{n-1}'(x)\)

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

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	1														1
2	0	1	2													3
3	0	0	3	4												7
4	0	0	2	10	9											21
5	0	0	0	10	30	21										61
6	0	0	0	5	45	90	51									191
7	0	0	0	0	35	175	266	127								603
8	0	0	0	0	14	196	644	784	323							1961
9	0	0	0	0	0	126	924	2268	2304	835						6457
10	0	0	0	0	0	42	840	3990	7770	6765	2188					21595
11	0	0	0	0	0	0	462	4620	16170	26070	19855	5798				72975
12	0	0	0	0	0	0	132	3564	22770	62700	86130	58278	15511			249085
13	0	0	0	0	0	0	0	1716	22308	103818	235092	281138	171106	41835		857013
14	0	0	0	0	0	0	0	429	15015	123123	447447	858858	908908	502593	113634	2970007

Technical details

Node id: n-188454bd1b8bfd31 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-0ea939b55f107842

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, 1;
  2: 0, 1, 2;
  3: 0, 0, 3,  4;
  4: 0, 0, 2, 10,  9;
  5: 0, 0, 0, 10, 30,  21;
  6: 0, 0, 0,  5, 45,  90,  51;
  7: 0, 0, 0,  0, 35, 175, 266,  127;
  8: 0, 0, 0,  0, 14, 196, 644,  784,  323;
  9: 0, 0, 0,  0,  0, 126, 924, 2268, 2304, 835;

%F Formula

The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = -(4*n-4+(17*n-20)*x+(28*n-34)*x^2+(21*n-24)*x^3+(6*n-6)*x^4)*P(n-1,x) + (8*x+30*x^2+42*x^3+26*x^4+6*x^5)*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[(-(4*n - 4 + (17*n - 20)*x + (28*n - 34)*x^2 + (21*n - 24)*x^3 + (6*n - 6)*x^4)*P[n - 1, x] + (8*x + 30*x^2 + 42*x^3 + 26*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (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; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n - 4 + (17*n - 20)*x + (28*n - 34)*x^2 + (21*n - 24)*x^3 + (6*n - 6)*x^4)*v[i - 1] + (8*x + 30*x^2 + 42*x^3 + 26*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (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 A Motzkin related triangle.?
  - 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