A358735 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-6\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 12\,n - 8\,n x + 2\,n x^{2} - 6\,n^{2} + 12\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 10\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 5 + 8\,x + 2\,x^{2}\)

\(P_{3}(x) = 11 + 42\,x + 34\,x^{2} + 6\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	5	8	2													15
3	11	42	34	6												93
4	21	160	298	172	24											675
5	36	483	1712	2208	1028	120										5587
6	57	1232	7368	17492	17848	7128	720									51845
7	85	2772	25764	100172	183768	158728	56376	5040								532705
8	121	5664	77088	454084	1335128	2037688	1550624	501408	40320							6002125
9	166	10725	204336	1724736	7539196	18142840	24072256	16563024	4956192	362880						73576351
10	221	19096	491634	5702268	35136584	123889152	255885488	303755680	192405024	53916480	3628800					974830427
11	287	32318	1092806	16855826	140698144	690390528	2065234064	3777115232	4092402080	2417838624	640255680	39916800				13881832389
12	365	52416	2273986	45430216	498020304	3273047040	13485420464	35384234592	58564585888	58764143616	32704996608	8241903360	479001600			211443110455
13	456	81991	4474912	113307584	1591148184	13595777712	74335851936	266363224624	627523525024	954988721920	897243125568	474029339904	114329767680	6227020800		3430345368295
14	561	124320	8394464	264526392	4661985616	50571517200	356521163664	1682288132448	5373672753152	11562900895680	16375971747584	14530043750016	7331890254336	1700249080320	87178291200	59056222616953

Technical details

Node id: n-b0fa9c2f4eae3fec · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6b585dfe392e21e2

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:   2,     1;
  2:   5,     8,      2;
  3:  11,    42,     34,       6;
  4:  21,   160,    298,     172,      24;
  5:  36,   483,   1712,    2208,    1028,      120;
  6:  57,  1232,   7368,   17492,   17848,     7128,      720;
  7:  85,  2772,  25764,  100172,  183768,   158728,    56376,     5040;
  8: 121,  5664,  77088,  454084, 1335128,  2037688,  1550624,   501408,   40320;
  9: 166, 10725, 204336, 1724736, 7539196, 18142840, 24072256, 16563024, 4956192, 362880;

%F Formula

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 5 + 8*x + 2*x^2, P(3,x) = 11 + 42*x + 34*x^2 + 6*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 5 + 8*x + 2*x^2;
P[3, x_] := 11 + 42*x + 34*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
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] = 2 + x; v[3] = 5 + 8*x + 2*x^2; v[4] = 11 + 42*x + 34*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
  - 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
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

References

[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