A327997 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-54 + 45\,n - 9\,n^{2}\right)\,P_{n-4}(x) + \left(-6 - 24\,x - 9\,n + 12\,n x + 6\,n^{2}\right)\,P_{n-3}(x) + \left(24\,x^{2} - 12\,n x^{2}\right)\,P_{n-3}'(x) + \left(8 - 4\,x + 9\,x^{2} - 8\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(4\,x^{2} + 4\,x^{3} + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(3 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 9 + 28\,x + 9\,x^{2}\)

\(P_{3}(x) = 27 + 152\,x + 108\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	9	28	9						46
3	27	152	108	16					303
4	81	768	909	288	25				2071
5	243	3876	6795	3440	625	36			15015
6	729	20460	49446	35280	10000	1188	49		117152
7	2187	117288	368739	343504	133875	24444	2058	64	992159

All 15 rows

Technical details

Node id: n-9017368722c57371 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-c3bf6a1d4d14368e

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:     3,       4;
  2:     9,      28,        9;
  3:    27,     152,      108,       16;
  4:    81,     768,      909,      288,       25;
  5:   243,    3876,     6795,     3440,      625,      36;
  6:   729,   20460,    49446,    35280,    10000,    1188,      49;
  7:  2187,  117288,   368739,   343504,   133875,   24444,    2058,     64;
  8:  6561,  751968,  2912922,  3349696,  1665475,  412272,   52822,   3328,   81;
  9: 19683, 5482980, 24884334, 33781088, 20341125, 6346620, 1092798, 104064, 5103, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-9*n-6+(12*n-24)*x)*P(n-3,x) - (12*n-24)*x^2*d/dx P(n-3,x) - (n^2+8*n-8+(4*n+4)*x-9*x^2)*P(n-2,x) + ((4*n+4)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n+3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 9 + 28*x + 9*x^2, P(3,x) = 27 + 152*x + 108*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 9 + 28*x + 9*x^2;
P[3, x_] := 27 + 152*x + 108*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 9*n - 6 + (12*n - 24)*x)*P[n - 3, x] - (12*n - 24)*x^2*D[P[n - 3, x], x] - (n^2 + 8*n - 8 + (4*n + 4)*x - 9*x^2)*P[n - 2, x] + ((4*n + 4)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n + 3 + 4*x)*P[n - 1, x] - 4*x^2*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] = 3 + 4*x; v[3] = 9 + 28*x + 9*x^2; v[4] = 27 + 152*x + 108*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 9*n - 6 + (12*n - 24)*x)*v[i - 3] - (12*n - 24)*x^2*deriv(v[i - 3], x) - (n^2 + 8*n - 8 + (4*n + 4)*x - 9*x^2)*v[i - 2] + ((4*n + 4)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n + 3 + 4*x)*v[i - 1] - 4*x^2*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 Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - 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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023