Real-rooted atlas

A052553 → x ↦ x+1

nonnegative coefficients

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

Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-5 - 4\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3213
4325
555111
6810321
713209143
8213822485

Show all 15 stored rows

Generated b-files: A128100, A283642 (row sums) ?

Computed in your browser ?

Other recurrences

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

\(P_{n}(x) = \left(1 + x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
Technical details
Node id
n-622af52a4165c278 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-cfa07bd223f21b32)
Fingerprint
f6-c12a5875baa63cc4

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1;
   3:  2,   1;
   4:  3,   2;
   5:  5,   5,   1;
   6:  8,  10,   3;
   7: 13,  20,   9,  1;
   8: 21,  38,  22,  4;
   9: 34,  71,  51, 14, 1;
  10: 55, 130, 111, 40, 5;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (3*n-5+(2*n-4)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 5 + (2*n - 4)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n - 5 + (2*n - 4)*x)*v[i - 1] - (5 + 9*x + 4*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A128100: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  1;
  2:  2,   1;
  3:  3,   2;
  4:  5,   5,   1;
  5:  8,  10,   3;
  6: 13,  20,   9,  1;
  7: 21,  38,  22,  4;
  8: 34,  71,  51, 14, 1;
  9: 55, 130, 111, 40, 5;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n-2+(2*n-2)*x)*P(n-1,x) - (5+9*x+4*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[((3*n - 2 + (2*n - 2)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = ((3*n - 2 + (2*n - 2)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*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 >= 1, 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-13fa50edf65b735e
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-13fa50edf65b735e, e-bb56aa5752b1a607
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-13fa50edf65b735e, e-bb56aa5752b1a607
  • Real-rooted. Every row P_n, n >= 1, 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-bb56aa5752b1a607
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-bb56aa5752b1a607

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

Same fingerprint (4)

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

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