Real-rooted atlas

A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

A091042 nonnegative coefficients

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

Recurrence from OEIS A091042
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 + 2\,x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
3110516
412135764
5136126849256
6155330462165111024
717871517161287286134096
8110513655005643530034551516384

Show all 15 stored rows

Generated b-files: A000302 (row sums), A091042, A103327 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-ba8de4e02acfb6ca · revision 1 · JSON
Definition source
oeis_catalog:A091042
Fingerprint
f6-49c3a6b95fc852bf

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,   3;
   3: 1,  10,    5;
   4: 1,  21,   35,     7;
   5: 1,  36,  126,    84,     9;
   6: 1,  55,  330,   462,   165,    11;
   7: 1,  78,  715,  1716,  1287,   286,    13;
   8: 1, 105, 1365,  5005,  6435,  3003,   455,    15;
   9: 1, 136, 2380, 12376, 24310, 19448,  6188,   680,  17;
  10: 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 19;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*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 + 3*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*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.

A091042: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,   3;
  2: 1,  10,    5;
  3: 1,  21,   35,     7;
  4: 1,  36,  126,    84,     9;
  5: 1,  55,  330,   462,   165,    11;
  6: 1,  78,  715,  1716,  1287,   286,    13;
  7: 1, 105, 1365,  5005,  6435,  3003,   455,    15;
  8: 1, 136, 2380, 12376, 24310, 19448,  6188,   680,  17;
  9: 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 19;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*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.

A103327: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  3,   1;
  2:  5,  10,     1;
  3:  7,  35,    21,     1;
  4:  9,  84,   126,    36,     1;
  5: 11, 165,   462,   330,    55,     1;
  6: 13, 286,  1287,  1716,   715,    78,     1;
  7: 15, 455,  3003,  6435,  5005,  1365,   105,    1;
  8: 17, 680,  6188, 19448, 24310, 12376,  2380,  136,   1;
  9: 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];
Table[Reverse[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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-d98f7e0074b379c4
  • 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-1c0df90858517049
  • 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-1c0df90858517049
  • 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-1c0df90858517049
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-1c0df90858517049

Proof chains

2 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 14 proof steps

Same fingerprint (4)

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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023