Real-rooted atlas

A145597 → basis A008277

nonnegative coefficients

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

Recurrence
\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(-2\,x - 2\,x^{2} - 3\,n x - 3\,n x^{2} - n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 4\,x^{2} - 3\,n x - 6\,n x^{2} - n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} - 3\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(20 + 10\,x + 18\,n + 9\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(10\,x + 9\,n x + 2\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
22733666
311025010510475
442916321194255153525
5163898911125640535252127384
661885754095872518421110296628223538
723256326880768348586026184590262081638361916982

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-16dc5705b09e1e7c · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b282ad822797bc0a

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:      6,        3;
  2:     27,       33,        6;
  3:    110,      250,      105,       10;
  4:    429,     1632,     1194,      255,       15;
  5:   1638,     9891,    11256,     4053,      525,      21;
  6:   6188,    57540,    95872,    51842,    11102,     966,      28;
  7:  23256,   326880,   768348,   586026,   184590,   26208,    1638,     36;
  8:  87210,  1831095,  5923665,  6128280,  2664045,  548730,   55440,   2610,   45;
  9: 326876, 10172965, 44514921, 60831463, 35065470, 9826113, 1426194, 107778, 3960, 55;
%F Formula
The row polynomials satisfy (n^2+6*n)*P(n,x) = -((n^2+3*n+2)*x+(n^2+3*n+2)*x^2)*P(n-2,x) - ((n^2+3*n+2)*x+(2*n^2+6*n+4)*x^2)*d/dx P(n-2,x) - (n^2+3*n+2)*x^2*d^2/dx^2 P(n-2,x) + (4*n^2+18*n+20+(2*n^2+9*n+10)*x)*P(n-1,x) + (2*n^2+9*n+10)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^2 + 3*n + 2)*x + (n^2 + 3*n + 2)*x^2)*P[n - 2, x] - ((n^2 + 3*n + 2)*x + (2*n^2 + 6*n + 4)*x^2)*D[P[n - 2, x], x] - (n^2 + 3*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (4*n^2 + 18*n + 20 + (2*n^2 + 9*n + 10)*x)*P[n - 1, x] + (2*n^2 + 9*n + 10)*x*D[P[n - 1, x], x])/(n^2 + 6*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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n^2 + 3*n + 2)*x + (n^2 + 3*n + 2)*x^2)*v[i - 2] - ((n^2 + 3*n + 2)*x + (2*n^2 + 6*n + 4)*x^2)*deriv(v[i - 2], x) - (n^2 + 3*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n^2 + 18*n + 20 + (2*n^2 + 9*n + 10)*x)*v[i - 1] + (2*n^2 + 9*n + 10)*x*deriv(v[i - 1], x))/(n^2 + 6*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-a01aa1ae1565a201
  • 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-a01aa1ae1565a201
  • 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-a01aa1ae1565a201

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

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023