A145597 → γ

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(-\frac{6}{5}\,n - \frac{1}{5}\,n^{2}\right)\,P_{n}(x) = \left(-2 - \frac{9}{5}\,n - \frac{2}{5}\,n^{2}\right)\,P_{n-1}(x) + \left(\frac{2}{5} - \frac{8}{5}\,x + \frac{3}{5}\,n - \frac{12}{5}\,n x + \frac{1}{5}\,n^{2} - \frac{4}{5}\,n^{2} x\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	3				3
2	6	3			9
3	10	15			25
4	15	45	9		69
5	21	105	63		189
6	28	210	252	28	518
7	36	378	756	252	1422

Show all 15 stored rows

Generated b-files: A005322 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-92efd0b8c392ac45 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-5b6b8f74397d2d43

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

OEIS matches ?

A005322: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 2 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  3;
  2:  6,   3;
  3: 10,  15;
  4: 15,  45,    9;
  5: 21, 105,   63;
  6: 28, 210,  252,   28;
  7: 36, 378,  756,  252;
  8: 45, 630, 1890, 1260,  90;
  9: 55, 990, 4158, 4620, 990;

%F Formula

The row polynomials satisfy -((1/5)*n^2+(6/5)*n)*P(n,x) = -((2/5)*n^2+(9/5)*n+2)*P(n-1,x) + ((1/5)*n^2+(3/5)*n+2/5-((4/5)*n^2+(12/5)*n+8/5)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-((2/5)*n^2 + (9/5)*n + 2)*P[n - 1, x] + ((1/5)*n^2 + (3/5)*n + 2/5 - ((4/5)*n^2 + (12/5)*n + 8/5)*x)*P[n - 2, x])/(-(1/5)*n^2 - (6/5)*n)]];
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; for(n = 2, N, my(i = n + 1); v[i] = (-((2/5)*n^2 + (9/5)*n + 2)*v[i - 1] + ((1/5)*n^2 + (3/5)*n + 2/5 - ((4/5)*n^2 + (12/5)*n + 8/5)*x)*v[i - 2])/(-(1/5)*n^2 - (6/5)*n)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-fbbe2b618491014f
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-fbbe2b618491014f
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-fbbe2b618491014f

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

gamma_polynomial scale 1, shift 0 e-fbbe2b618491014f

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-bcb2e3135d28c083 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-e8333902099cfe55
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-bcb2e3135d28c083 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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