A075498 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 10\,x + 10\,x^{2} + 3\,x^{3} - 3\,n x - 6\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	9	27	19													55
3	27	144	225	109												505
4	81	729	1926	2019	742											5497
5	243	3726	14904	25371	19764	5815										69823
6	729	19683	111861	274644	339921	209376	51193									1007407
7	2187	107892	836163	2776761	4855059	4696182	2385543	498118								16157905
8	6561	610173	6292728	27198261	63060201	85195557	67451238	29103057	5296321							284214097
9	19683	3529818	47934666	263223675	778237389	1377786915	1511709156	1010933163	378560493	60987817						5432922775
10	59049	20726199	370873647	2546687142	9350814474	20840375709	29716839846	27381171960	15823481004	5228047857	754940848					112034017735
11	177147	122821920	2921567373	24808407273	110987752446	302944703832	539100479070	641710889865	508730660316	258527470158	76357501956	9983845261				2476196276617
12	531441	731794257	23463356346	244349584335	1313041487616	4302664132194	9293311962936	13721760041274	13991462210727	9721175972364	4404408375654	1175336169084	140329768789			58332035387017
13	1594323	4373759430	192124424556	2438778674619	15570827443170	60346183701519	155041227254484	276139542066246	347707464769521	309672273923442	191313801259866	78144952854060	19007840477703	2087182244308		1457666574447247
14	4782969	26186755275	1602218618937	24689026991160	185763068168679	841788844513902	2533722512870979	5334081828007932	8065212570896625	8839853502514605	6981516674578161	3879996047161506	1441975105767483	322082034811587	32725315072135	38485034941511935

Technical details

Node id: n-f656951313aec62a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-90c6c54322d6b2c6

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:     3,       4;
  2:     9,      27,       19;
  3:    27,     144,      225,       109;
  4:    81,     729,     1926,      2019,       742;
  5:   243,    3726,    14904,     25371,     19764,       5815;
  6:   729,   19683,   111861,    274644,    339921,     209376,      51193;
  7:  2187,  107892,   836163,   2776761,   4855059,    4696182,    2385543,     498118;
  8:  6561,  610173,  6292728,  27198261,  63060201,   85195557,   67451238,   29103057,   5296321;
  9: 19683, 3529818, 47934666, 263223675, 778237389, 1377786915, 1511709156, 1010933163, 378560493, 60987817;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (3-(3*n-10)*x-(6*n-10)*x^2-(3*n-3)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+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[((3 - (3*n - 10)*x - (6*n - 10)*x^2 - (3*n - 3)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = ((3 - (3*n - 10)*x - (6*n - 10)*x^2 - (3*n - 3)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + x)); 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 Stirling2 triangle with scaled diagonals (powers of 3).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.