A144698 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 11\,x^{2} + 5\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	5														6
2	1	15	30													46
3	1	35	180	210												426
4	1	75	750	2100	1680											4606
5	1	155	2700	13650	25200	15120										56826
6	1	315	9030	73500	235200	317520	151200									786766
7	1	635	28980	357210	1764000	4021920	4233600	1663200								12069546
8	1	1275	90750	1631700	11677680	40007520	69854400	59875200	19958400							203096926
9	1	2555	279900	7162050	71442000	345144240	889056000	1247400000	898128000	259459200						3718073946
10	1	5115	855030	30607500	414506400	2713843440	9674834400	19758816000	23051952000	14270256000	3632428800					73548104686
11	1	10235	2595780	128415210	2317392000	20013618240	94862275200	264493706400	444573360000	442377936000	239740300800	54486432000				1562996041866
12	1	20475	7848750	531831300	12614281680	140938237440	864172108800	3159434678400	7175084716800	10203233040000	8830434412800	4249941696000	871782912000			35508175784446
13	1	40955	23669100	2182266450	67326058800	959157959760	7458587136000	34781370624000	102488978592000	195308431718400	239980041100800	183455816544000	79332244992000	14820309504000		858654470206266
14	1	81915	71253030	8894749500	354088425600	6360882287760	61801689549600	360295423488000	1339777254816000	3285441038880000	5374098496166400	5801170415040000	3966612249600000	1556132497920000	266765571072000	22018818573329806

Technical details

Node id: n-51dc433de0f0e26a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-2a9069f1f4217ff4

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: 1,    5;
  2: 1,   15,     30;
  3: 1,   35,    180,     210;
  4: 1,   75,    750,    2100,     1680;
  5: 1,  155,   2700,   13650,    25200,     15120;
  6: 1,  315,   9030,   73500,   235200,    317520,    151200;
  7: 1,  635,  28980,  357210,  1764000,   4021920,   4233600,    1663200;
  8: 1, 1275,  90750, 1631700, 11677680,  40007520,  69854400,   59875200,  19958400;
  9: 1, 2555, 279900, 7162050, 71442000, 345144240, 889056000, 1247400000, 898128000, 259459200;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+7*x+11*x^2+5*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[((1 + 7*x + 11*x^2 + 5*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] = ((1 + 7*x + 11*x^2 + 5*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 Triangle of 4-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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