Real-rooted atlas

A108838: Triangle of Dyck paths counted by number of long interior inclines.

A108838 nonnegative coefficients

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

Recurrence from OEIS A108838
\(\left(3 + 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(6 + 8\,n + 8\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(3 - 6\,x + 3\,x^{2} - n + 2\,n x - n x^{2} - 2\,n^{2} + 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1325
248214
352015242
464060242132
5770175140352429
681124205602804821430
79168882176414705046324862

Show all 15 stored rows

Generated b-files: A108838 ?

Computed in your browser ?

Technical details
Node id
n-c963abfc0266d66c · revision 1 · JSON
Definition source
oeis_catalog:A108838
Fingerprint
f6-4e2cf4a8bda0ea40

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:  2;
  1:  3,   2;
  2:  4,   8,    2;
  3:  5,  20,   15,     2;
  4:  6,  40,   60,    24,     2;
  5:  7,  70,  175,   140,    35,     2;
  6:  8, 112,  420,   560,   280,    48,    2;
  7:  9, 168,  882,  1764,  1470,   504,   63,    2;
  8: 10, 240, 1680,  4704,  5880,  3360,  840,   80,  2;
  9: 11, 330, 2970, 11088, 19404, 16632, 6930, 1320, 99, 2;
%F Formula
The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = (4*n^2+8*n+6+(4*n^2+8*n)*x)*P(n-1,x) - (2*n^2+n-3-(4*n^2+2*n-6)*x+(2*n^2+n-3)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 8*n + 6 + (4*n^2 + 8*n)*x)*P[n - 1, x] - (2*n^2 + n - 3 - (4*n^2 + 2*n - 6)*x + (2*n^2 + n - 3)*x^2)*P[n - 2, x])/(2*n^2 + 7*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 8*n + 6 + (4*n^2 + 8*n)*x)*v[i - 1] - (2*n^2 + n - 3 - (4*n^2 + 2*n - 6)*x + (2*n^2 + n - 3)*x^2)*v[i - 2])/(2*n^2 + 7*n + 3)); 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.

A108838: triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2:  2;
   3:  3,   2;
   4:  4,   8,    2;
   5:  5,  20,   15,     2;
   6:  6,  40,   60,    24,     2;
   7:  7,  70,  175,   140,    35,     2;
   8:  8, 112,  420,   560,   280,    48,    2;
   9:  9, 168,  882,  1764,  1470,   504,   63,    2;
  10: 10, 240, 1680,  4704,  5880,  3360,  840,   80,  2;
  11: 11, 330, 2970, 11088, 19404, 16632, 6930, 1320, 99, 2;
%F Formula
The row polynomials satisfy (2*n^2-n-3)*P(n,x) = (4*n^2-8*n+6+(4*n^2-8*n)*x)*P(n-1,x) - (2*n^2-7*n+3-(4*n^2-14*n+6)*x+(2*n^2-7*n+3)*x^2)*P(n-2,x) for n >= 4, with P(2,x) = 2, P(3,x) = 3 + 2*x.
%t Mathematica
P[2, x_] := 2;
P[3, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + 6 + (4*n^2 - 8*n)*x)*P[n - 1, x] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*P[n - 2, x])/(2*n^2 - n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 4, N, my(i = n - 1); v[i] = ((4*n^2 - 8*n + 6 + (4*n^2 - 8*n)*x)*v[i - 1] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*v[i - 2])/(2*n^2 - n - 3)); 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(11)

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-0cfae28328f92be1
  • 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-a4549745e8cd4154
  • 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-a4549745e8cd4154
  • 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-a4549745e8cd4154
  • 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-a4549745e8cd4154

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 (7)

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