A079638 → reverse

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	1	9	14													24
3	1	18	83	90												192
4	1	30	275	870	744											1920
5	1	45	685	4275	10474	7560										23040
6	1	63	1435	14805	70924	143892	91440									322560
7	1	84	2674	41160	324289	1274196	2233356	1285200								5160960
8	1	108	4578	98280	1151409	7398972	24870572	38769840	20603520							92897280
9	1	135	7350	209790	3419913	32559975	177945200	526791060	745785936	371226240						1857945600
10	1	165	11220	410850	8889573	117466965	947490830	4533332100	12072814776	15766233120	7428153600					40874803200
11	1	198	16445	751410	20830623	364669074	4071519815	28637334990	122524830076	298275087528	363640940640	163459296000				980995276800
12	1	234	23309	1299870	44919303	1006107102	14852759207	144200769570	903331310596	3512369295864	7914320990784	9092247615360	3923502105600			25505877196800
13	1	273	32123	2147145	90483393	2523519999	47591781809	609681627555	5256847141546	29807441892228	106668976545768	224688938947200	245065185624960	102017281766400		714164561510400
14	1	315	43225	3411135	172179007	5850669825	137228995475	2242313478405	25457071147508	198105941713680	1029768938962400	3426283530174960	6801187780761984	7085176946023680	2856571067750400	21424936845312000

Technical details

Node id: n-fd3aab150239f748 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-bd19e0a95c26bd62

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: 1,   3;
  2: 1,   9,   14;
  3: 1,  18,   83,     90;
  4: 1,  30,  275,    870,     744;
  5: 1,  45,  685,   4275,   10474,     7560;
  6: 1,  63, 1435,  14805,   70924,   143892,     91440;
  7: 1,  84, 2674,  41160,  324289,  1274196,   2233356,   1285200;
  8: 1, 108, 4578,  98280, 1151409,  7398972,  24870572,  38769840,  20603520;
  9: 1, 135, 7350, 209790, 3419913, 32559975, 177945200, 526791060, 745785936, 371226240;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 2*n)*x^2*P[n - 2, x] + (1 + 3*n*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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 2*n)*x^2*v[i - 2] + (1 + 3*n*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.?
  - 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

Related ?

Same fingerprint ?

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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023