A111006 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(2 + 19\,x + 63\,x^{2} + 95\,x^{3} + 67\,x^{4} + 18\,x^{5} - 2\,n - 18\,n x - 60\,n x^{2} - 92\,n x^{3} - 66\,n x^{4} - 18\,n x^{5}\right)\,P_{n-1}(x) + \left(4\,x + 29\,x^{2} + 81\,x^{3} + 109\,x^{4} + 71\,x^{5} + 18\,x^{6}\right)\,P_{n-1}'(x)\)

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

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	3													4
3	0	0	2	5												7
4	0	0	1	7	11											19
5	0	0	0	3	16	21										40
6	0	0	0	1	12	41	43									97
7	0	0	0	0	4	34	94	85								217
8	0	0	0	0	1	18	99	219	171							508
9	0	0	0	0	0	5	60	261	492	341						1159
10	0	0	0	0	0	1	25	195	678	1101	683					2683
11	0	0	0	0	0	0	6	95	576	1692	2426	1365				6160
12	0	0	0	0	0	0	1	33	340	1644	4149	5311	2731			14209
13	0	0	0	0	0	0	0	7	140	1106	4488	9959	11528	5461		32689
14	0	0	0	0	0	0	0	1	42	546	3430	11925	23568	24881	10923	75316

Technical details

Node id: n-a25d5a5440f7c216 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-3d8013fbbe82cb22

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, 3;
  3: 0, 0, 2, 5;
  4: 0, 0, 1, 7, 11;
  5: 0, 0, 0, 3, 16, 21;
  6: 0, 0, 0, 1, 12, 41, 43;
  7: 0, 0, 0, 0,  4, 34, 94,  85;
  8: 0, 0, 0, 0,  1, 18, 99, 219, 171;
  9: 0, 0, 0, 0,  0,  5, 60, 261, 492, 341;

%F Formula

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = -(2*n-2+(18*n-19)*x+(60*n-63)*x^2+(92*n-95)*x^3+(66*n-67)*x^4+(18*n-18)*x^5)*P(n-1,x) + (4*x+29*x^2+81*x^3+109*x^4+71*x^5+18*x^6)*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[(-(2*n - 2 + (18*n - 19)*x + (60*n - 63)*x^2 + (92*n - 95)*x^3 + (66*n - 67)*x^4 + (18*n - 18)*x^5)*P[n - 1, x] + (4*x + 29*x^2 + 81*x^3 + 109*x^4 + 71*x^5 + 18*x^6)*D[P[n - 1, x], x])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = (-(2*n - 2 + (18*n - 19)*x + (60*n - 63)*x^2 + (92*n - 95)*x^3 + (66*n - 67)*x^4 + (18*n - 18)*x^5)*v[i - 1] + (4*x + 29*x^2 + 81*x^3 + 109*x^4 + 71*x^5 + 18*x^6)*deriv(v[i - 1], x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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 Another version of Fibonacci-Pascal triangle A037027.?
  - 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