A249247 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-4}(x) + \left(4\,x - 2\,n x\right)\,P_{n-3}(x) + \left(-2 + 2\,x + 3\,n - n^{2}\right)\,P_{n-2}(x) + \left(-2 + 2\,n\right)\,P_{n-1}(x)\)

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

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

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

\(P_{3}(x) = 6 + 20\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	2	6							8
3	6	20	3						29
4	24	84	21						129
5	120	432	135	4					691
6	720	2640	936	52					4348
7	5040	18720	7200	544	5				31509
8	40320	151200	61560	5600	105				258785
9	362880	1370880	582120	60000	1625	6			2377511
10	3628800	13789440	6048000	682080	23250	186			24171756
11	39916800	152409600	68584320	8279040	330750	3996	7		269524513
12	479001600	1836172800	843696000	107412480	4821600	75852	301		3271180633
13	6227020800	23950080000	11196662400	1487808000	73029600	1382976	8575	8	42935992359
14	87178291200	336259123200	159507532800	21954240000	1156680000	25147584	208544	456	606081223784

Technical details

Node id: n-b1efee3e909d775c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3ad075d3fea555f2

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      1,       2;
  2:      2,       6;
  3:      6,      20,      3;
  4:     24,      84,     21;
  5:    120,     432,    135,     4;
  6:    720,    2640,    936,    52;
  7:   5040,   18720,   7200,   544,    5;
  8:  40320,  151200,  61560,  5600,  105;
  9: 362880, 1370880, 582120, 60000, 1625, 6;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-4)*x*P(n-3,x) - (n^2-3*n+2-2*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 2 + 6*x, P(3,x) = 6 + 20*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 2 + 6*x;
P[3, x_] := 6 + 20*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 4)*x*P[n - 3, x] - (n^2 - 3*n + 2 - 2*x)*P[n - 2, x] + (2*n - 2)*P[n - 1, 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 + 2*x; v[3] = 2 + 6*x; v[4] = 6 + 20*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 4)*x*v[i - 3] - (n^2 - 3*n + 2 - 2*x)*v[i - 2] + (2*n - 2)*v[i - 1]); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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.