Real-rooted atlas

A030528 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + n\right)\,P_{n}(x) = \left(6 + 2\,x - 2\,n\right)\,P_{n-1}(x) + \left(5 + 6\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
23418
35106122
4822218160
513455936101164
6218814712455121448
734167339366225781411224

Show all 15 stored rows

Generated b-files: A028859 (row sums), A154929 ?

Computed in your browser ?

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + x\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Technical details
Node id
n-51a2369b25b8e3ba · revision 1 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-9efbd63f6346959f)
Fingerprint
f6-1d84b6f5e9f3dbf6

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 ?

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:  2,   1;
  2:  3,   4,    1;
  3:  5,  10,    6,    1;
  4:  8,  22,   21,    8,    1;
  5: 13,  45,   59,   36,   10,    1;
  6: 21,  88,  147,  124,   55,   12,   1;
  7: 34, 167,  339,  366,  225,   78,  14,   1;
  8: 55, 310,  741,  976,  770,  370, 105,  16,  1;
  9: 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(2*n-6-2*x)*P(n-1,x) + (5+6*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 6 - 2*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 6 - 2*x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(n + 1)); 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.

A154929: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   4,    1;
  3:  5,  10,    6,    1;
  4:  8,  22,   21,    8,    1;
  5: 13,  45,   59,   36,   10,    1;
  6: 21,  88,  147,  124,   55,   12,   1;
  7: 34, 167,  339,  366,  225,   78,  14,   1;
  8: 55, 310,  741,  976,  770,  370, 105,  16,  1;
  9: 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(2*n-6-2*x)*P(n-1,x) + (5+6*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 6 - 2*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 6 - 2*x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f5ed3e64bc4c2256
  • 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-cd3fa89de2aead53, e-f5ed3e64bc4c2256
  • 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-cd3fa89de2aead53, e-f5ed3e64bc4c2256
  • 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-cd3fa89de2aead53

Proof chains

3 incoming proof steps. 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  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