Real-rooted atlas

A091044: One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

A091044 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A091044
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 + 2\,x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2310316
342828464
4560126605256
5611039639611061024
6718210011716100118274096
782802184572057202184280816384

Show all 15 stored rows

Generated b-files: A000302 (row sums), A091044, A091044 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(1 - x + n + 3\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-2245b35ec7cd4857 · revision 2 · JSON
Definition source
oeis_catalog:A091044
derived:reversal (equal by e-07d02f4f92ce5b13)
Fingerprint
f6-722f205416540a1b

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: yes
  • γ-positive (gamma_positive): yes
  • 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;
  2:  3,  10,    3;
  3:  4,  28,   28,     4;
  4:  5,  60,  126,    60,     5;
  5:  6, 110,  396,   396,   110,     6;
  6:  7, 182, 1001,  1716,  1001,   182,     7;
  7:  8, 280, 2184,  5720,  5720,  2184,   280,    8;
  8:  9, 408, 4284, 15912, 24310, 15912,  4284,  408,   9;
  9: 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10;
%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) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*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, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A091044: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,  10,    3;
   4:  4,  28,   28,     4;
   5:  5,  60,  126,    60,     5;
   6:  6, 110,  396,   396,   110,     6;
   7:  7, 182, 1001,  1716,  1001,   182,     7;
   8:  8, 280, 2184,  5720,  5720,  2184,   280,    8;
   9:  9, 408, 4284, 15912, 24310, 15912,  4284,  408,   9;
  10: 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10;
%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) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*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], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*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 = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A091044: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,  10,    3;
   4:  4,  28,   28,     4;
   5:  5,  60,  126,    60,     5;
   6:  6, 110,  396,   396,   110,     6;
   7:  7, 182, 1001,  1716,  1001,   182,     7;
   8:  8, 280, 2184,  5720,  5720,  2184,   280,    8;
   9:  9, 408, 4284, 15912, 24310, 15912,  4284,  408,   9;
  10: 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10;
%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 >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*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]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*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 = concat(c, vector(n - #c)); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e55cd7ac2c40d500
  • Real-rooted. Every row P_n, n >= 0, 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-4ffbabf9b60188a8
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-4ffbabf9b60188a8, e-75f779345b39f910
  • Hadamard kernel (Malo). Rows P_n, n >= 0, 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-4ffbabf9b60188a8, e-75f779345b39f910
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-4ffbabf9b60188a8
  • Real-rooted. Every row P_n, n >= 0, 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-75f779345b39f910

Proof chains

4 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 16 proof steps

Same fingerprint (2)

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