A075504 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-162 + 63\,x + 8\,x^{2}\right)\,P_{n-2}(x) + \left(-81\,x + 72\,x^{2}\right)\,P_{n-2}'(x) + \left(27 - 7\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 108 + 29\,x + x^{2}\)

\(P_{1}(x) = 1296 + 675\,x + 57\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	108	29	1								138
1	1296	675	57	1							2029
2	17496	14985	2295	94	1						34871
3	262440	334611	81405	5805	140	1					684402
4	4251528	7669809	2740311	300510	12285	195	1				14974639
5	72275976	181575675	90522117	14222061	873180	23058	259	1			359492327
6	1262703816	4434875145	2984041215	641232774	55250181	2153466	39690	332	1		9380296620
7	22384294920	111247075971	98999486685	28169030205	3259898460	175487067	4711770	63990	414	1	264240049483

All 15 rows

Technical details

Node id: n-aba3f426cbb4ac99 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-bcfe4e830a38fffc

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:           108,             29,               1;
  1:          1296,            675,              57,              1;
  2:         17496,          14985,            2295,             94,              1;
  3:        262440,         334611,           81405,           5805,            140,            1;
  4:       4251528,        7669809,         2740311,         300510,          12285,          195,           1;
  5:      72275976,      181575675,        90522117,       14222061,         873180,        23058,         259,          1;
  6:    1262703816,     4434875145,      2984041215,      641232774,       55250181,      2153466,       39690,        332,        1;
  7:   22384294920,   111247075971,     98999486685,    28169030205,     3259898460,    175487067,     4711770,      63990,      414,      1;
  8:  399817944648,  2850080350689,   3318227790471,  1221607483470,   184115077245,  13117440105,   480442347,    9408960,    98010,    505,   1;
  9: 7168828728456, 74181338131275, 112491284045877, 52713622692621, 10114842151260, 925725580956, 44350739433, 1173283947, 17481420, 144045, 605, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(162-63*x-8*x^2)*P(n-2,x) - (81*x-72*x^2)*d/dx P(n-2,x) + (27-7*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 108 + 29*x + x^2, P(1,x) = 1296 + 675*x + 57*x^2 + x^3.

%t Mathematica

P[0, x_] := 108 + 29*x + x^2;
P[1, x_] := 1296 + 675*x + 57*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(162 - 63*x - 8*x^2)*P[n - 2, x] - (81*x - 72*x^2)*D[P[n - 2, x], x] + (27 - 7*x)*P[n - 1, x] + 9*x*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] = 108 + 29*x + x^2; v[2] = 1296 + 675*x + 57*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(162 - 63*x - 8*x^2)*v[i - 2] - (81*x - 72*x^2)*deriv(v[i - 2], x) + (27 - 7*x)*v[i - 1] + 9*x*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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Stirling2 triangle with scaled diagonals (powers of 9).?
  - 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.