A008459 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(x + x^{2} - n x - n x^{2}\right)\,P_{n-2}(x) + \left(x + 2\,x^{2} - n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(-2 - x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-x + 2\,n x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	6	7	1						14
3	20	43	15	1					79
4	70	251	157	26	1				505
5	252	1431	1415	415	40	1			3554
6	924	8065	11791	5340	905	57	1		27083
7	3432	45207	93821	60991	15750	1736	77	1	221015

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ae235faa570528c4 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-b334f8e538f872f3

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:     1;
  1:     2,       1;
  2:     6,       7,       1;
  3:    20,      43,      15,       1;
  4:    70,     251,     157,      26,       1;
  5:   252,    1431,    1415,     415,      40,      1;
  6:   924,    8065,   11791,    5340,     905,     57,     1;
  7:  3432,   45207,   93821,   60991,   15750,   1736,    77,    1;
  8: 12870,  252859,  725173,  647010,  235851,  39270,  3038,  100,   1;
  9: 48620, 1413943, 5499519, 6535327, 3208212, 745941, 86646, 4962, 126, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n - 1)*x + (n - 1)*x^2)*P[n - 2, x] - ((n - 1)*x + (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x])/(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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-((n - 1)*x + (n - 1)*x^2)*v[i - 2] - ((n - 1)*x + (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 2 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x))/(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-906bba3237df75f5
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-906bba3237df75f5
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-906bba3237df75f5

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-906bba3237df75f5

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Square the entries of Pascal's triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-c6b04c4560617585 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-1906182487ade7a7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square the entries of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-c6b04c4560617585 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A069466 → basis A008277

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