A358735 → reverse

nonnegative coefficients

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

Recurrence

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

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

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

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	1														2
2	2	4	1													7
3	6	16	10	1												33
4	24	76	70	20	1											191
5	120	428	496	224	35	1										1304
6	720	2808	3808	2260	588	56	1									10241
7	5040	21096	32152	23008	8140	1344	84	1								90865
8	40320	178848	298688	245560	107328	24772	2772	120	1							898409
9	362880	1690272	3040848	2786320	1424920	414596	66352	5280	165	1						9791634
10	3628800	17628480	33748704	33766048	19523152	6750240	1385384	160732	9438	220	1					116601199
11	39916800	201170880	406129824	437233664	279245920	110512624	27314784	4124224	358930	16016	286	1				1506023953
12	479001600	2493884160	5272270848	6041435136	4191669664	1850877280	529279504	97370016	11170224	749320	26026	364	1			20967734143
13	6227020800	33378497280	73487372544	88882027584	66168446080	32007510304	10303142192	2213856592	312638160	27957384	1478048	40768	455	1		313009988192
14	87178291200	479753003520	1095101208576	1388829247104	1098850029440	574400601280	204067341632	49749399392	8275367920	919309680	65437216	2777528	61880	560	1	4987192076929

Technical details

Node id: n-40ebc8eb92a30e0e · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-b3267034d12950ac

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,       1;
  2:      2,       4,       1;
  3:      6,      16,      10,       1;
  4:     24,      76,      70,      20,       1;
  5:    120,     428,     496,     224,      35,      1;
  6:    720,    2808,    3808,    2260,     588,     56,     1;
  7:   5040,   21096,   32152,   23008,    8140,   1344,    84,    1;
  8:  40320,  178848,  298688,  245560,  107328,  24772,  2772,  120,   1;
  9: 362880, 1690272, 3040848, 2786320, 1424920, 414596, 66352, 5280, 165, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] + (n + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] + (n + 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 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 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

Related ?

Same fingerprint ?

A358735: 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(…

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