Real-rooted atlas

A113214: Riordan array (1+2x,x(1+x)).

A113214 nonnegative coefficients

Proved for all n: all roots real, in (−4, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A113214
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
20314
302417
40055111
500296118
60007147129
7000216208147

Show all 15 stored rows

Generated b-files: A000204 (row sums), A113214 ?

Computed in your browser ?

Other recurrences

Basis-map coefficients of A113214, shifted · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(4 + x - 2\,n\right)\,P_{n-1}(x) + \left(4\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-c27b11147fce8bfa · revision 1 · JSON
Definition source
oeis_catalog:A113214
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A113214 (equal by e-738c0b3c1f38e8ef)
Fingerprint
f6-088313a0be276231

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: 0, 3, 1;
  3: 0, 2, 4, 1;
  4: 0, 0, 5, 5,  1;
  5: 0, 0, 2, 9,  6,  1;
  6: 0, 0, 0, 7, 14,  7,  1;
  7: 0, 0, 0, 2, 16, 20,  8,  1;
  8: 0, 0, 0, 0,  9, 30, 27,  9,  1;
  9: 0, 0, 0, 0,  2, 25, 50, 35, 10, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,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[x*P[n - 1, x] + x*P[n - 2, x]];
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] = x*v[i - 1] + x*v[i - 2]); 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.

A113214: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 2, 1;
  2: 0, 3, 1;
  3: 0, 2, 4, 1;
  4: 0, 0, 5, 5,  1;
  5: 0, 0, 2, 9,  6,  1;
  6: 0, 0, 0, 7, 14,  7,  1;
  7: 0, 0, 0, 2, 16, 20,  8,  1;
  8: 0, 0, 0, 0,  9, 30, 27,  9,  1;
  9: 0, 0, 0, 0,  2, 25, 50, 35, 10, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,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[x*P[n - 1, x] + x*P[n - 2, x]];
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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*v[i - 2]); 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

Proof chains

15 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

Used in 26 proof steps

Same fingerprint (1)

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

References

  1. [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