A358735 → basis A007318 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

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

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	2														3
2	2	8	7													17
3	6	34	60	33												133
4	24	172	442	484	191											1313
5	120	1028	3408	5480	4283	1304										15623
6	720	7128	28648	59972	69096	41604	10241									217409
7	5040	56376	264568	676608	1020012	907612	441840	90865								3462921
8	40320	501408	2679584	8051416	14897528	17401044	12541744	5103716	898409							62115169
9	362880	4956192	29626704	101881792	222378760	319805596	303267168	182985016	63784717	9791634						1238840459
10	3628800	53916480	355701024	1373836960	3443689520	5858549664	6855441304	5451655132	2821125618	858229228	116601199					27192374929
11	39916800	640255680	4613262624	19731363680	55710838496	109110128272	151348621216	148769137808	101600631346	45932169526	12375808436	1506023953				651378157837
12	479001600	8241903360	64319102208	301308124416	944413452448	2087899441248	3340281560880	3898251801056	3295120064160	1968143218024	788738948978	190475376312	20967734143			16908639728833
13	6227020800	114329767680	959736962304	4881151894848	16792383485440	41289163159264	74686441071664	100672386698656	101158989228704	74863893658904	39674153975728	14261331957200	3117335380103	313009988192		472790534249487
14	87178291200	1700249080320	15265114753536	83683676021376	313127480011520	846425111872320	1705292372099072	2602308571486560	3023643754819472	2662740087082864	1749965334138496	832476967794424	271035267687664	54069030540024	4987192076929	14166807387755777

Technical details

Node id: n-f325e9d4bddfdf8c · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-cfc761a86542e28e

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,       2;
  2:      2,       8,        7;
  3:      6,      34,       60,        33;
  4:     24,     172,      442,       484,       191;
  5:    120,    1028,     3408,      5480,      4283,      1304;
  6:    720,    7128,    28648,     59972,     69096,     41604,     10241;
  7:   5040,   56376,   264568,    676608,   1020012,    907612,    441840,     90865;
  8:  40320,  501408,  2679584,   8051416,  14897528,  17401044,  12541744,   5103716,   898409;
  9: 362880, 4956192, 29626704, 101881792, 222378760, 319805596, 303267168, 182985016, 63784717, 9791634;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] + (n + (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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] + (n + (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 (-1, 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 basis_map_A007318 of 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…?
  - 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

Related ?

Same fingerprint ?

A358735 → basis A007318

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023