A249253 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3 + 2\,x\)

\(P_{2}(x) = 12 + 10\,x\)

\(P_{3}(x) = 60 + 56\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	12	10							22
3	60	56	3						119
4	360	360	33						753
5	2520	2640	315	4					5479
6	20160	21840	3060	76					45136
7	181440	201600	31500	1104	5				415649
8	1814400	2056320	347760	15120	145				4233745
9	19958400	22982400	4127760	208320	2975	6			47279861
10	239500800	279417600	52617600	2963520	54600	246			574554366
11	3113510400	3672345600	718502400	44029440	970200	6768	7		7549364815
12	43589145600	51891840000	10478160000	686568960	17287200	160272	385		106663162417
13	653837184000	784604620800	162680918400	11256537600	314344800	3568320	13671	8	1612697187599
14	10461394944000	12640852224000	2680732454400	194075481600	5887728000	77837760	405720	568	25983021076048

Technical details

Node id: n-b8b5d35fdd4f7bb6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-808052d03351391b

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:        3,        2;
  2:       12,       10;
  3:       60,       56,       3;
  4:      360,      360,      33;
  5:     2520,     2640,     315,      4;
  6:    20160,    21840,    3060,     76;
  7:   181440,   201600,   31500,   1104,    5;
  8:  1814400,  2056320,  347760,  15120,  145;
  9: 19958400, 22982400, 4127760, 208320, 2975, 6;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*n*x*P(n-3,x) - (n^2+n-2*x)*P(n-2,x) + (2*n+2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 12 + 10*x, P(3,x) = 60 + 56*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 12 + 10*x;
P[3, x_] := 60 + 56*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*n*x*P[n - 3, x] - (n^2 + n - 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] = 3 + 2*x; v[3] = 12 + 10*x; v[4] = 60 + 56*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 2*n*x*v[i - 3] - (n^2 + n - 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.