A282629 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 12\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 9\,x^{2}\right)\,P_{n-2}'(x) + \left(-1 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	6														7
2	1	30	27													58
3	1	126	324	108												559
4	1	510	2835	2376	405											6127
5	1	2046	22140	35100	14175	1458										74920
6	1	8190	164187	439560	315900	74358	5103									1007299
7	1	32766	1186164	5052348	5755050	2326968	357210	17496								14728003
8	1	131070	8450595	55268136	93761955	57949668	14931378	1609632	59049							232161484
9	1	524286	59743980	586483740	1426160925	1264737726	486520020	86605200	6908733	196830						3917881441
10	1	2097150	420567147	6103813320	20739406050	25369982946	13670462421	3573383040	465010875	28540350	649539					70373912839
11	1	8388606	2953407204	62720401788	292501578600	480581588916	348533726310	125484583752	23685439635	2349166050	114318864	2125764				1338934725490
12	1	33554430	20711599155	639017646696	4037722028505	8742311105616	8304176361096	3955633618464	1015646461038	144728114850	11296132749	446410440	6908733			26871729941773
13	1	134217726	145132189020	6473022863580	54886702545675	154412776992474	188377439730480	115495401415680	38741424988266	7437875419260	827782894224	52147116684	1706457051	22320522		566851569150643
14	1	536870910	1016529302907	65310757391880	737800968832200	2668196561044014	4119616074352779	3186764338792320	1358332604484570	337398595033500	50206258604502	4482109621080	232617040110	6405989814	71744535	12529364429105122

Technical details

Node id: n-c5b24384f32713ef · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3444b903bbab1a5c

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: 1,      6;
  2: 1,     30,       27;
  3: 1,    126,      324,       108;
  4: 1,    510,     2835,      2376,        405;
  5: 1,   2046,    22140,     35100,      14175,       1458;
  6: 1,   8190,   164187,    439560,     315900,      74358,      5103;
  7: 1,  32766,  1186164,   5052348,    5755050,    2326968,    357210,    17496;
  8: 1, 131070,  8450595,  55268136,   93761955,   57949668,  14931378,  1609632,   59049;
  9: 1, 524286, 59743980, 586483740, 1426160925, 1264737726, 486520020, 86605200, 6908733, 196830;

%F Formula

The row polynomials satisfy P(n,x) = (2+12*x-9*x^2)*P(n-2,x) + (6*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 >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[(2 + 12*x - 9*x^2)*P[n - 2, x] + (6*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, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 12*x - 9*x^2)*v[i - 2] + (6*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 - 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 Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].?
  - 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.