A225466 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	6							8
3	4	42	27						73
4	8	234	405	108					755
5	16	1218	4293	2808	405				8740
6	32	6186	39825	48060	16200	1458			111761
7	64	31122	346437	687960	407025	83106	5103		1560817
8	128	155994	2911545	8953308	8278200	2878092	392931	17496	23587694

All 15 rows

Technical details

Node id: n-db679d4b6f6e5c94 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b925348db8e9e89d

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   2,       6;
   3:   4,      42,        27;
   4:   8,     234,       405,        108;
   5:  16,    1218,      4293,       2808,        405;
   6:  32,    6186,     39825,      48060,      16200,       1458;
   7:  64,   31122,    346437,     687960,     407025,      83106,      5103;
   8: 128,  155994,   2911545,    8953308,    8278200,    2878092,    392931,     17496;
   9: 256,  780738,  23994333,  110132568,  149469705,   78729084,  17931942,   1749600,   59049;
  10: 512, 3905226, 195467985, 1307435580, 2505573000, 1876485366, 634190634, 101721744, 7440174, 196830;

%F Formula

The row polynomials satisfy P(n,x) = (2+6*x-9*x^2)*P(n-2,x) + (3*x-9*x^2)*d/dx P(n-2,x) + (1+6*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[(2 + 6*x - 9*x^2)*P[n - 2, x] + (3*x - 9*x^2)*D[P[n - 2, x], x] + (1 + 6*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = (2 + 6*x - 9*x^2)*v[i - 2] + (3*x - 9*x^2)*deriv(v[i - 2], x) + (1 + 6*x)*v[i - 1] + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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 Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…?
  - 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.