A367270 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2\,x - 2\,x^{2} + 5\,n x + 5\,n x^{2} - 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 4\,x^{2} + 5\,n x + 10\,n x^{2} - 2\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 5\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(6 + 4\,x - 16\,n - 8\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - 8\,n x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	3	2						5
3	10	15	3					28
4	35	94	42	4				175
5	126	555	425	90	5			1201
6	462	3191	3765	1340	165	6		8929
7	1716	18109	31087	16590	3395	273	7	71177

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0138853f54741ca2 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-579ccb35e2b28c68

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:     1;
  2:     3,      2;
  3:    10,     15,       3;
  4:    35,     94,      42,       4;
  5:   126,    555,     425,      90,      5;
  6:   462,   3191,    3765,    1340,    165,      6;
  7:  1716,  18109,   31087,   16590,   3395,    273,     7;
  8:  6435, 102110,  246162,  185108,  55930,   7420,   420,   8;
  9: 24310, 574107, 1898157, 1934562, 808899, 156870, 14574, 612, 9;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -((2*n^2-5*n+2)*x+(2*n^2-5*n+2)*x^2)*P(n-2,x) - ((2*n^2-5*n+2)*x+(4*n^2-10*n+4)*x^2)*d/dx P(n-2,x) - (2*n^2-5*n+2)*x^2*d^2/dx^2 P(n-2,x) + (8*n^2-16*n+6+(4*n^2-8*n+4)*x)*P(n-1,x) + (4*n^2-8*n+4)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((2*n^2 - 5*n + 2)*x + (2*n^2 - 5*n + 2)*x^2)*P[n - 2, x] - ((2*n^2 - 5*n + 2)*x + (4*n^2 - 10*n + 4)*x^2)*D[P[n - 2, x], x] - (2*n^2 - 5*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + (4*n^2 - 8*n + 4)*x*D[P[n - 1, x], x])/(2*n^2 - 3*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-((2*n^2 - 5*n + 2)*x + (2*n^2 - 5*n + 2)*x^2)*v[i - 2] - ((2*n^2 - 5*n + 2)*x + (4*n^2 - 10*n + 4)*x^2)*deriv(v[i - 2], x) - (2*n^2 - 5*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*v[i - 1] + (4*n^2 - 8*n + 4)*x*deriv(v[i - 1], x))/(2*n^2 - 3*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-f7f7b35ffa4336f4
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-f7f7b35ffa4336f4
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-f7f7b35ffa4336f4

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-f7f7b35ffa4336f4

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-2b43df6c63e34146 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