Real-rooted atlas

A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

A033282 nonnegative coefficients

Proved for all n: all roots real, in (−1, 0) · interlacing · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A033282
\(\left(2 + n\right)\,P_{n}(x) = \left(1 + 2\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x) + \left(1 - n\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
215511
319211445
4114568442197
5120120300330132903
6127225825148512874294279
71353851925500570075005143020793

Show all 15 stored rows

Generated b-files: A033282, A126216 (reversed triangle) ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3} + n + 4\,n x + 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

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

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 4\,x + n + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-66e2c3448245b0b7 · revision 2 · JSON
Definition source
oeis_catalog:A033282
derived:mobius_x_over_x_plus_one (equal by e-3b43508d22a3c256)
derived:reversal (equal by e-594b6f87d0c18695)
Fingerprint
f6-1e5fbd8fe1b88630

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: 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: 1,  2;
  2: 1,  5,   5;
  3: 1,  9,  21,   14;
  4: 1, 14,  56,   84,    42;
  5: 1, 20, 120,  300,   330,   132;
  6: 1, 27, 225,  825,  1485,  1287,    429;
  7: 1, 35, 385, 1925,  5005,  7007,   5005,   1430;
  8: 1, 44, 616, 4004, 14014, 28028,  32032,  19448,  4862;
  9: 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+1+(4*n+2)*x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (4*n + 2)*x)*P[n - 1, x] - (n - 1)*P[n - 2, x])/(n + 2)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1 + (4*n + 2)*x)*v[i - 1] - (n - 1)*v[i - 2])/(n + 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.

A033282: triangle, offset 3, from the row P0
%e Example
Triangle begins:
   3: 1;
   4: 1,  2;
   5: 1,  5,   5;
   6: 1,  9,  21,   14;
   7: 1, 14,  56,   84,    42;
   8: 1, 20, 120,  300,   330,   132;
   9: 1, 27, 225,  825,  1485,  1287,    429;
  10: 1, 35, 385, 1925,  5005,  7007,   5005,   1430;
  11: 1, 44, 616, 4004, 14014, 28028,  32032,  19448,  4862;
  12: 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n-5+(4*n-10)*x)*P(n-1,x) - (n-4)*P(n-2,x) for n >= 5, with P(3,x) = 1, P(4,x) = 1 + 2*x.
%t Mathematica
P[3, x_] := 1;
P[4, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 5 + (4*n - 10)*x)*P[n - 1, x] - (n - 4)*P[n - 2, x])/(n - 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 2)); v[1] = 1; v[2] = 1 + 2*x; for(n = 5, N, my(i = n - 2); v[i] = ((2*n - 5 + (4*n - 10)*x)*v[i - 1] - (n - 4)*v[i - 2])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

Generated with polytool c48629374597.

A126216: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:     1;
   2:     2,     1;
   3:     5,     5,      1;
   4:    14,    21,      9,      1;
   5:    42,    84,     56,     14,     1;
   6:   132,   330,    300,    120,    20,     1;
   7:   429,  1287,   1485,    825,   225,    27,    1;
   8:  1430,  5005,   7007,   5005,  1925,   385,   35,   1;
   9:  4862, 19448,  32032,  28028, 14014,  4004,  616,  44,  1;
  10: 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (2*n-1+(4*n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n + 1)]];
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] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n + 1)); 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-484268c09e1d1880
  • 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-484268c09e1d1880, e-5bc6f2a826329589, e-7909c64e1764eda7
  • 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-7909c64e1764eda7
  • 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-3adfaf26de79e7ad, e-5bc6f2a826329589, e-7909c64e1764eda7, e-e5f3b61b305da328
  • 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-3adfaf26de79e7ad, e-5bc6f2a826329589, e-7909c64e1764eda7, e-e5f3b61b305da328
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-1, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5bc6f2a826329589
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e5f3b61b305da328
  • 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-3adfaf26de79e7ad

Proof chains

7 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 (6)

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