A075504 → θ+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 81 + 54\,x + 3\,x^{2}\)

\(P_{1}(x) = 729 + 1134\,x + 162\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	81	54	3								138
1	729	1134	162	4							2029
2	6561	21870	6075	360	5						34871
3	59049	406782	196830	21060	675	6					684402
4	531441	7440174	5924583	1020600	56700	1134	7				14974639
5	4782969	134986014	171124002	44641044	3827250	129276	1764	8			359492327
6	43046721	2439314190	4822827075	1835242920	228027555	11573604	261954	2592	9		9380296620
7	387420489	43993748862	133875302310	72499181220	12555293625	898607682	30005640	486000	3645	10	264240049483

All 15 rows

Technical details

Node id: n-955637f4530b33f6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-bbb5df5595f8ddbd

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:          81,             54,               3;
  1:         729,           1134,             162,               4;
  2:        6561,          21870,            6075,             360,              5;
  3:       59049,         406782,          196830,           21060,            675,             6;
  4:      531441,        7440174,         5924583,         1020600,          56700,          1134,            7;
  5:     4782969,      134986014,       171124002,        44641044,        3827250,        129276,         1764,          8;
  6:    43046721,     2439314190,      4822827075,      1835242920,      228027555,      11573604,       261954,       2592,         9;
  7:   387420489,    43993748862,    133875302310,     72499181220,    12555293625,     898607682,     30005640,     486000,      3645,      10;
  8:  3486784401,   792662320494,   3680623785663,   2788470927000,   655612189650,   63591167178,   2938730949,   69284160,    841995,    4950,   11;
  9: 31381059609, 14274895337694, 100565835693642, 105292451752884, 32988137193000, 4220657655192, 259329744828, 8347009176, 146146275, 1381050, 6534, 12;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 9*x^2*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 81 + 54*x + 3*x^2, P(1,x) = 729 + 1134*x + 162*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 81 + 54*x + 3*x^2;
P[1, x_] := 729 + 1134*x + 162*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 9*x^2*D[P[n - 2, x], x] + (9 + 2*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] = 81 + 54*x + 3*x^2; v[2] = 729 + 1134*x + 162*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 9*x^2*deriv(v[i - 2], x) + (9 + 2*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.