A144697 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)

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

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	4														5
2	1	12	20													33
3	1	28	120	120												269
4	1	60	500	1200	840											2601
5	1	124	1800	7800	12600	6720										29045
6	1	252	6020	42000	117600	141120	60480									367473
7	1	508	19320	204120	882000	1787520	1693440	604800								5191709
8	1	1020	60500	932400	5838840	17781120	27941760	21772800	6652800							80981241
9	1	2044	186600	4092600	35721000	153397440	355622400	453600000	299376000	79833600						1381831685
10	1	4092	570020	17490000	207253200	1206152640	3869933760	7185024000	7683984000	4390848000	1037836800					25599096513
11	1	8188	1730520	73380120	1158696000	8894941440	37944910080	96179529600	148191120000	136116288000	68497228800	14529715200				511587547949
12	1	16380	5232500	303903600	6307140840	62639216640	345668843520	1148885337600	2391694905600	3139456320000	2522981260800	1133317785600	217945728000			10969205691081
13	1	32764	15779400	1247009400	33663029400	426292426560	2983434854400	12647771136000	34162992864000	60094902067200	68565726028800	48921551078400	19833061248000	3487131648000		251157789202325
14	1	65532	47502020	5082714000	177044212800	2827058794560	24720675819840	131016517632000	446592418272000	1010904935040000	1535456713190400	1546978777344000	991653062400000	366148823040000	59281238016000	6115762394043153

Technical details

Node id: n-adb48c983f7b4925 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-f5092c6face8a76f

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,    4;
  2: 1,   12,     20;
  3: 1,   28,    120,     120;
  4: 1,   60,    500,    1200,      840;
  5: 1,  124,   1800,    7800,    12600,      6720;
  6: 1,  252,   6020,   42000,   117600,    141120,     60480;
  7: 1,  508,  19320,  204120,   882000,   1787520,   1693440,    604800;
  8: 1, 1020,  60500,  932400,  5838840,  17781120,  27941760,  21772800,   6652800;
  9: 1, 2044, 186600, 4092600, 35721000, 153397440, 355622400, 453600000, 299376000, 79833600;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+6*x+9*x^2+4*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 + 6*x + 9*x^2 + 4*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 + 6*x + 9*x^2 + 4*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 3-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