A108838 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(3 + 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(3\,x + 3\,x^{2} - n x - n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3\,x + 6\,x^{2} - n x - 2\,n x^{2} - 2\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(3\,x^{2} - n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(6 + 16\,n + 8\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,n x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	5	2							7
2	14	14	2						30
3	42	79	27	2					150
4	132	418	254	44	2				850
5	429	2162	2070	620	65	2			5348
6	1430	11106	15702	7180	1280	90	2		36790
7	4862	57059	114471	74676	19950	2359	119	2	273498

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d1a6f682aa835d49 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-91c41abeecd96e75

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

OEIS matches ?

No OEIS matches.

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:     2;
  1:     5,       2;
  2:    14,      14,       2;
  3:    42,      79,      27,       2;
  4:   132,     418,     254,      44,       2;
  5:   429,    2162,    2070,     620,      65,      2;
  6:  1430,   11106,   15702,    7180,    1280,     90,      2;
  7:  4862,   57059,  114471,   74676,   19950,   2359,    119,    2;
  8: 16796,  294158,  815474,  727524,  271782,  47572,   4004,  152,   2;
  9: 58786, 1524005, 5730663, 6794468, 3398283, 820890, 101388, 6384, 189, 2;

%F Formula

The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = -((2*n^2+n-3)*x+(2*n^2+n-3)*x^2)*P(n-2,x) - ((2*n^2+n-3)*x+(4*n^2+2*n-6)*x^2)*d/dx P(n-2,x) - (2*n^2+n-3)*x^2*d^2/dx^2 P(n-2,x) + (8*n^2+16*n+6+(4*n^2+8*n)*x)*P(n-1,x) + (4*n^2+8*n)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 2*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((2*n^2 + n - 3)*x + (2*n^2 + n - 3)*x^2)*P[n - 2, x] - ((2*n^2 + n - 3)*x + (4*n^2 + 2*n - 6)*x^2)*D[P[n - 2, x], x] - (2*n^2 + n - 3)*x^2*D[P[n - 2, x], {x, 2}] + (8*n^2 + 16*n + 6 + (4*n^2 + 8*n)*x)*P[n - 1, x] + (4*n^2 + 8*n)*x*D[P[n - 1, x], x])/(2*n^2 + 7*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-((2*n^2 + n - 3)*x + (2*n^2 + n - 3)*x^2)*v[i - 2] - ((2*n^2 + n - 3)*x + (4*n^2 + 2*n - 6)*x^2)*deriv(v[i - 2], x) - (2*n^2 + n - 3)*x^2*deriv(deriv(v[i - 2], x), x) + (8*n^2 + 16*n + 6 + (4*n^2 + 8*n)*x)*v[i - 1] + (4*n^2 + 8*n)*x*deriv(v[i - 1], x))/(2*n^2 + 7*n + 3)); 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-b4a3a1de2e82786d
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-b4a3a1de2e82786d
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-b4a3a1de2e82786d

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

basis_map A008277 e-b4a3a1de2e82786d

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a4549745e8cd4154 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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.
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023