Real-rooted atlas

A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.

A011973 → x ↦ (1/4)x−1/4

A034867 A202064 nonnegative coefficients

Proved for all n: all roots real, in (−∞, 1] what this means

Recurrence from OEIS A034867
\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(-1 + x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2314
3448
4510116
5620632
673521164
7856568128
8984126361256
91012025212010512
10111654623305511024
1112220792792220122048
1213286128717167157814096
1314364200234322002364148192
14154553003643550051365105116384

Generated b-files: A000079 (row sums), A034867, A109447 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-fd133d9d2402ed18 · revision 1 · JSON
Definition source
oeis_catalog:A034867
Fingerprint
f6-937db0732045a112

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 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;
  2:  3,   1;
  3:  4,   4;
  4:  5,  10,   1;
  5:  6,  20,   6;
  6:  7,  35,  21,   1;
  7:  8,  56,  56,   8;
  8:  9,  84, 126,  36,  1;
  9: 10, 120, 252, 120, 10;
%F Formula
The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - 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; for(n = 2, N, my(i = n + 1); v[i] = 2*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A034867: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  2;
  2:  3,   1;
  3:  4,   4;
  4:  5,  10,   1;
  5:  6,  20,   6;
  6:  7,  35,  21,   1;
  7:  8,  56,  56,   8;
  8:  9,  84, 126,  36,  1;
  9: 10, 120, 252, 120, 10;
%F Formula
The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - 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; for(n = 2, N, my(i = n + 1); v[i] = 2*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A109447: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2;
   3:  1,   3;
   4:  4,   4;
   5:  1,  10,   5;
   6:  6,  20,   6;
   7:  1,  21,  35,   7;
   8:  8,  56,  56,   8;
   9:  1,  36, 126,  84,  9;
  10: 10, 120, 252, 120, 10;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-e41ac5ad698885bc

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

Same fingerprint (11)

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