Real-rooted atlas

A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.

A063967 nonnegative coefficients

Proved for all n: all roots real, in (−5, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A063967
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
32316
4375116
5515167144
6830432991120
713581049546111328
82110923527117967131896

Show all 15 stored rows

Generated b-files: A063967 ?

Computed in your browser ?

Other recurrences

Basis-map coefficients of A063967, shifted · proved equal for all n (proof step)

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

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

\(P_{n}(x) = \left(1 + x\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
Technical details
Node id
n-c72653d9ab0ee8be · revision 2 · JSON
Definition source
oeis_catalog:A063967
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A063967 (equal by e-4cad2095b142c753)
derived:basis_map_A007318 (equal by e-d339e2ece1e12101)
Fingerprint
f6-4d750740fbfda5f4

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 -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,   1;
   3:  2,   3,    1;
   4:  3,   7,    5,    1;
   5:  5,  15,   16,    7,    1;
   6:  8,  30,   43,   29,    9,    1;
   7: 13,  58,  104,   95,   46,   11,   1;
   8: 21, 109,  235,  271,  179,   67,  13,   1;
   9: 34, 201,  506,  705,  591,  303,  92,  15,  1;
  10: 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 1] + (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, v[i])))};
T(10)

Generated with polytool c48629374597.

A063967: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  1,   1;
  2:  2,   3,    1;
  3:  3,   7,    5,    1;
  4:  5,  15,   16,    7,    1;
  5:  8,  30,   43,   29,    9,    1;
  6: 13,  58,  104,   95,   46,   11,   1;
  7: 21, 109,  235,  271,  179,   67,  13,   1;
  8: 34, 201,  506,  705,  591,  303,  92,  15,  1;
  9: 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (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

18 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 25 proof steps

Same fingerprint (1)

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