A285061 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 24\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(12\,x - 16\,x^{2}\right)\,P_{n-2}'(x) + \left(-2 + 8\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x\)

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	8														9
2	1	48	48													97
3	1	248	720	256												1225
4	1	1248	7968	7168	1280											17665
5	1	6248	79200	135680	57600	6144										284873
6	1	31248	750288	2186240	1657600	405504	28672									5059553
7	1	156248	6940080	32422656	39110400	16472064	2609152	131072								97841673
8	1	781248	63398208	458508288	826990080	533643264	142098432	15728640	589824							2041737985
9	1	3906248	575271360	6298731520	16351372800	15176060928	6042796032	1105723392	90243072	2621440						45646726793
10	1	19531248	5200879728	84951623680	309466995200	397183868928	221891518464	59690188800	7953776640	498073600	11534336					1086867990625
11	1	97656248	46925105040	1132109133056	5685697036800	9826302824448	7400812683264	2745376702464	531080478720	53778841600	2664431616	50331648				27424895224905
12	1	488281248	422911882848	14967685956608	102317395290880	233643705090048	230876396929024	113448210923520	29879850958848	4350174822400	345868599296	13891534848	218103808			730266798373377
13	1	2441406248	3809136633120	196835447477760	1814234149728000	5397641304287232	6862247213645824	4345433074171904	1496552030797824	293755806023680	33321381789696	2134363865088	70883737600	939524096		20446038173088073
14	1	12207031248	34296878135568	2579176212587520	31826157782764800	122058791308726272	196745173094486016	157387832127651840	68940665850101760	17520307181977600	2658702199881728	241448767193088	12722213224448	355140108288	4026531840	600005644990402017

Technical details

Node id: n-893812c1efc3542a · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-62e59b3741b0e3c5

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,       8;
  2: 1,      48,        48;
  3: 1,     248,       720,        256;
  4: 1,    1248,      7968,       7168,        1280;
  5: 1,    6248,     79200,     135680,       57600,        6144;
  6: 1,   31248,    750288,    2186240,     1657600,      405504,      28672;
  7: 1,  156248,   6940080,   32422656,    39110400,    16472064,    2609152,     131072;
  8: 1,  781248,  63398208,  458508288,   826990080,   533643264,  142098432,   15728640,   589824;
  9: 1, 3906248, 575271360, 6298731520, 16351372800, 15176060928, 6042796032, 1105723392, 90243072, 2621440;

%F Formula

The row polynomials satisfy P(n,x) = (3+24*x-16*x^2)*P(n-2,x) + (12*x-16*x^2)*d/dx P(n-2,x) - (2-8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[(3 + 24*x - 16*x^2)*P[n - 2, x] + (12*x - 16*x^2)*D[P[n - 2, x], x] - (2 - 8*x)*P[n - 1, x] + 4*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 24*x - 16*x^2)*v[i - 2] + (12*x - 16*x^2)*deriv(v[i - 2], x) - (2 - 8*x)*v[i - 1] + 4*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 S2[4,1] = (exp(x), exp(4*x) - 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.