A161198 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 20\,n - 8\,n x - 4\,n^{2}\right)\,P_{n-2}(x) + \left(-8 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	3	16	12													31
4	15	92	108	32												247
5	105	704	1032	512	80											2433
6	945	6756	11400	7360	2000	192										28653
7	10395	78096	145668	111360	40400	6912	448									393279
8	135135	1056828	2127972	1836128	803600	186816	21952	1024								6169455
9	2027025	16392960	35090064	33216512	16644320	4730880	765184	65536	2304							108934785
10	34459425	286788420	645709968	658254208	365994720	120371328	24046848	2863104	186624	5120						2138679765
11	654729075	5586817680	13128854652	14228723200	8599535200	3165442560	737756544	109363200	9987840	512000	11264					46221733215
12	13749310575	119942087580	292466400732	333813466272	216162047200	87113178240	22878920064	3982927872	455811840	32947200	1362944	24576				1090598485095
13	316234143225	2813665256640	7086553479576	8457620126208	5806260751280	2522392012800	729479244032	143902015488	19445260032	1770700800	103831552	3538944	53248			27897430413825
14	7905853580625	71606567988900	185604832759320	230337979100736	166300569097520	76994826123072	24122562464000	5264931516416	809911035648	87479208960	6491330560	315015168	8998912	114688		769042328334525
15	213458046676875	1965000750022800	5226150188468340	6714071656411392	5065960313384880	2477981671156992	830963780815168	197290436575232	33713693874432	4161740943360	367720184832	22668312576	925503488	22478848	245760	22729143615054975

Technical details

Node id: n-2b8d6f32cc61eaeb · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3f072313d028fa7f

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:        1,         4;
   3:        3,        16,        12;
   4:       15,        92,       108,        32;
   5:      105,       704,      1032,       512,        80;
   6:      945,      6756,     11400,      7360,      2000,       192;
   7:    10395,     78096,    145668,    111360,     40400,      6912,      448;
   8:   135135,   1056828,   2127972,   1836128,    803600,    186816,    21952,    1024;
   9:  2027025,  16392960,  35090064,  33216512,  16644320,   4730880,   765184,   65536,   2304;
  10: 34459425, 286788420, 645709968, 658254208, 365994720, 120371328, 24046848, 2863104, 186624, 5120;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-20*n+25+(8*n-20)*x+4*x^2)*P(n-2,x) + (4*n-8+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*P[n - 2, x] + (4*n - 8 + 4*x)*P[n - 1, 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] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*v[i - 2] + (4*n - 8 + 4*x)*v[i - 1]); 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 of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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.