A163936 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

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	1														1
2	0	1	3													4
3	0	1	10	15												26
4	0	1	25	105	105											236
5	0	1	56	490	1260	945										2752
6	0	1	119	1918	9450	17325	10395									39208
7	0	1	246	6825	56980	190575	270270	135135								660032
8	0	1	501	22935	302995	1636635	4099095	4729725	2027025							12818912
9	0	1	1012	74316	1487200	12122110	47507460	94594500	91891800	34459425						282137824
10	0	1	2035	235092	6914908	81431350	466876410	1422280860	2343240900	1964187225	654729075					6939897856
11	0	1	4082	731731	30950920	510880370	4104160060	17892864990	44346982680	62199262125	45831035250	13749310575				188666182784
12	0	1	8177	2252341	134779645	3049616570	33309926650	199124936010	694740296250	1446733012725	1764494857125	1159525191825	316234143225			5617349020544
13	0	1	16368	6879678	575156036	17539336815	254752658160	2026763158420	9540421090200	27610143335775	49473074851200	53338158823950	31623414322500	7905853580625		181790703209728
14	0	1	32751	20900922	2417578670	98049492723	1861763348445	19282395272140	118885395048420	458380554006375	1129764045234825	1774073543492250	1712934942468750	924984868933125	213458046676875	6353726042486272

Technical details

Node id: n-a5d4a3ad7140b153 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-014ae34d4ff839c2

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, 1,   10,    15;
  4: 0, 1,   25,   105,     105;
  5: 0, 1,   56,   490,    1260,      945;
  6: 0, 1,  119,  1918,    9450,    17325,    10395;
  7: 0, 1,  246,  6825,   56980,   190575,   270270,   135135;
  8: 0, 1,  501, 22935,  302995,  1636635,  4099095,  4729725,  2027025;
  9: 0, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n*x+2*n*x^2+n*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+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*x + 2*n*x^2 + n*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + 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*x + 2*n*x^2 + n*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*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 - 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 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    
    ↑ Liu–Wang theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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