A156366 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 9 + 12\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	9	12	1								22
1	27	99	33	1							160
2	81	702	594	78	1						1456
3	243	4617	8154	2718	171	1					15904
4	729	29160	96471	65232	10719	360	1				202672
5	2187	180063	1043199	1265139	421713	38637	741	1			2951680
6	6561	1097874	10649232	21440862	12651390	2382318	131472	1506	1		48361216
7	19683	6646293	104626080	331834968	318416022	106138674	12290184	430560	3039	1	880405504

All 15 rows

Technical details

Node id: n-9b2e23cf8b77cec6 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-c64977f436170997

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:      9,        12,          1;
  1:     27,        99,         33,           1;
  2:     81,       702,        594,          78,            1;
  3:    243,      4617,       8154,        2718,          171,            1;
  4:    729,     29160,      96471,       65232,        10719,          360,           1;
  5:   2187,    180063,    1043199,     1265139,       421713,        38637,         741,          1;
  6:   6561,   1097874,   10649232,    21440862,     12651390,      2382318,      131472,       1506,         1;
  7:  19683,   6646293,  104626080,   331834968,    318416022,    106138674,    12290184,     430560,      3039,       1;
  8:  59049,  40074588, 1001451357,  4819028256,   7099085106,   3820992264,   788787234,   59494176,   1373733,    6108,     1;
  9: 177147, 241097067, 9413808093, 66841401285, 145038502638, 118471456494, 39490485498, 5371796394, 275067495, 4304439, 12249, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3+(n+2)*x)*P(n-1,x) + (3*x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 9 + 12*x + x^2.

%t Mathematica

P[0, x_] := 9 + 12*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3 + (n + 2)*x)*P[n - 1, x] + (3*x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 12*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + (n + 2)*x)*v[i - 1] + (3*x - x^2)*deriv(v[i - 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 (−∞, 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 coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A156366: Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x)…

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