Real-rooted atlas

A008459: Square the entries of Pascal's triangle.

A008459 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 A008459
\(\left(n\right)\,P_{n}(x) = \left(-1 - x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(1 - 2\,x + x^{2} - n + 2\,n x - n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21416
3199120
41163616170
5125100100251252
6136225400225361924
7149441122512254414913432

Show all 15 stored rows

Generated b-files: A000984 (row sums), A008459, A008459 (reversed triangle) ?

Computed in your browser ?

Other recurrences

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

\(\left(n\right)\,P_{n}(x) = \left(-2\,x + n + 3\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-b13531ad79e7c629 · revision 2 · JSON
Definition source
oeis_catalog:A008459
derived:reversal (equal by e-ec2b4f4a19835fe0)
Fingerprint
f6-b8a804edad022635

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: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,    9,    1;
  4: 1, 16,   36,   16,     1;
  5: 1, 25,  100,  100,    25,     1;
  6: 1, 36,  225,  400,   225,    36,    1;
  7: 1, 49,  441, 1225,  1225,   441,   49,    1;
  8: 1, 64,  784, 3136,  4900,  3136,  784,   64,  1;
  9: 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-1-(2*n-2)*x+(n-1)*x^2)*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[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x])/(n)]];
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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*v[i - 2])/(n)); 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.

A008459: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,    9,    1;
  4: 1, 16,   36,   16,     1;
  5: 1, 25,  100,  100,    25,     1;
  6: 1, 36,  225,  400,   225,    36,    1;
  7: 1, 49,  441, 1225,  1225,   441,   49,    1;
  8: 1, 64,  784, 3136,  4900,  3136,  784,   64,  1;
  9: 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-1-(2*n-2)*x+(n-1)*x^2)*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[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x])/(n)]];
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] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*v[i - 2])/(n)); 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.

A008459: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,    9,    1;
  4: 1, 16,   36,   16,     1;
  5: 1, 25,  100,  100,    25,     1;
  6: 1, 36,  225,  400,   225,    36,    1;
  7: 1, 49,  441, 1225,  1225,   441,   49,    1;
  8: 1, 64,  784, 3136,  4900,  3136,  784,   64,  1;
  9: 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-1-(2*n-2)*x+(n-1)*x^2)*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[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x])/(n)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*v[i - 2])/(n)); 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 >= 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-2ee2b1274ddaaf7f
  • 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-c6b04c4560617585
  • 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-1906182487ade7a7, e-c6b04c4560617585
  • 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-1906182487ade7a7, e-c6b04c4560617585
  • 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-c6b04c4560617585
  • 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-1906182487ade7a7

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 14 proof steps

Same fingerprint (1)

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