Real-rooted atlas

narayana → D+1

nonnegative coefficients

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

Recurrence
\(\left(4 - 5\,x + x^{2} + n - 3\,n x + n x^{2} - 2\,n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(2 - 10\,x + 13\,x^{2} - 3\,x^{3} + n + 9\,n x - 17\,n x^{2} + 3\,n x^{3} - 8\,n^{2} x + 6\,n^{2} x^{2} + n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(3\,x - 9\,x^{2} + 8\,x^{3} - 2\,x^{4} - 3\,n x + 7\,n x^{2} - 4\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
345110
47189135
5115050141126
616115200110201462
7222316305952102711716
8294201666245014703643516435

Show all 15 stored rows

Generated b-files: A001700 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-085cc2fc18757071 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d3e13aaaafda97c1

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:  2,    1;
   3:  4,    5,    1;
   4:  7,   18,    9,     1;
   5: 11,   50,   50,    14,     1;
   6: 16,  115,  200,   110,    20,     1;
   7: 22,  231,  630,   595,   210,    27,    1;
   8: 29,  420, 1666,  2450,  1470,   364,   35,   1;
   9: 37,  708, 3864,  8232,  7644,  3192,  588,  44,  1;
  10: 46, 1125, 8100, 23688, 31752, 20412, 6300, 900, 54, 1;
%F Formula
The row polynomials satisfy (n^3-2*n^2+n+4+(2*n^2-3*n-5)*x+(n+1)*x^2)*P(n,x) = (n^3+n+2+(3*n^3-8*n^2+9*n-10)*x+(6*n^2-17*n+13)*x^2+(3*n-3)*x^3)*P(n-1,x) + ((2*n^2-3*n+3)*x-(2*n^2-7*n+9)*x^2-(4*n-8)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + n + 2 + (3*n^3 - 8*n^2 + 9*n - 10)*x + (6*n^2 - 17*n + 13)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + ((2*n^2 - 3*n + 3)*x - (2*n^2 - 7*n + 9)*x^2 - (4*n - 8)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + n + 4 + (2*n^2 - 3*n - 5)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^3 + n + 2 + (3*n^3 - 8*n^2 + 9*n - 10)*x + (6*n^2 - 17*n + 13)*x^2 + (3*n - 3)*x^3)*v[i - 1] + ((2*n^2 - 3*n + 3)*x - (2*n^2 - 7*n + 9)*x^2 - (4*n - 8)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + n + 4 + (2*n^2 - 3*n - 5)*x + (n + 1)*x^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.

Proved properties

  • 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-c4aee2e8ba50f587
  • 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-c4aee2e8ba50f587
  • 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-c4aee2e8ba50f587
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-c4aee2e8ba50f587

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023