A327997 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 9 + 14\,x + 3\,x^{2}\)

\(P_{3}(x) = 27 + 76\,x + 36\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	9	14	3													26
3	27	76	36	4												143
4	81	384	303	72	5											845
5	243	1938	2265	860	125	6										5437
6	729	10230	16482	8820	2000	198	7									38466
7	2187	58644	122913	85876	26775	4074	294	8								300771
8	6561	375984	970974	837424	333095	68712	7546	416	9							2600721
9	19683	2741490	8294778	8445272	4068225	1057770	156114	13008	567	10						24796917
10	59049	22785678	77433273	89792520	50381725	15719886	2903691	323280	21195	750	11					259421058
11	177147	214087212	791967726	1017188964	645156325	233083488	51404430	7130904	622215	33000	968	12				2960852391
12	531441	2245647888	8853380973	12333449304	8641077830	3518600184	895770953	147912336	16033842	1128600	49489	1224	13			36653584077
13	1594323	25978641570	107658348291	160187444820	121807351380	54757290588	15691038363	2985710728	384509268	33556380	1948947	71916	1521	14		489487508109
14	4782969	328081186134	1416219677064	2225539058652	1812149892735	885060063888	280005052327	59935405816	8885753019	920119200	66165099	3229044	101738	1862	15	7016870489562

Technical details

Node id: n-40abbfa038458015 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b76628d7fb7fc96d

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,       2;
  2:     9,      14,       3;
  3:    27,      76,      36,       4;
  4:    81,     384,     303,      72,       5;
  5:   243,    1938,    2265,     860,     125,       6;
  6:   729,   10230,   16482,    8820,    2000,     198,      7;
  7:  2187,   58644,  122913,   85876,   26775,    4074,    294,     8;
  8:  6561,  375984,  970974,  837424,  333095,   68712,   7546,   416,   9;
  9: 19683, 2741490, 8294778, 8445272, 4068225, 1057770, 156114, 13008, 567, 10;

%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+(6*n-12)*x)*P(n-3,x) - (n^2+8*n-8+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 9 + 14*x + 3*x^2, P(3,x) = 27 + 76*x + 36*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 9 + 14*x + 3*x^2;
P[3, x_] := 27 + 76*x + 36*x^2 + 4*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 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*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] = 9 + 14*x + 3*x^2; v[4] = 27 + 76*x + 36*x^2 + 4*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 + (6*n - 12)*x)*v[i - 3] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*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 - 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