Real-rooted atlas

narayana → γ → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(8 - 15\,x + 4\,x^{2} + 3\,n - 11\,n x + 4\,n x^{2} - 4\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(4 - 55\,x + 92\,x^{2} - 32\,x^{3} + n + 49\,n x - 72\,n x^{2} + 16\,n x^{3} - 2\,n^{2} - 24\,n^{2} x + 16\,n^{2} x^{2} + n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x - 38\,x^{2} + 64\,x^{3} - 32\,x^{4} - 6\,n x + 32\,n x^{2} - 32\,n x^{3} + 2\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3213
4437
5710219
611301051
71675455141
82216117535393

All 15 rows

Technical details
Node id
n-c9c651476acf61cd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2513bba9c35d0db8

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1;
   3:  2,   1;
   4:  4,   3;
   5:  7,  10,    2;
   6: 11,  30,   10;
   7: 16,  75,   45,   5;
   8: 22, 161,  175,  35;
   9: 29, 308,  560, 196,  14;
  10: 37, 540, 1512, 924, 126;
%F Formula
The row polynomials satisfy (n^3-4*n^2+3*n+8+(4*n^2-11*n-15)*x+(4*n+4)*x^2)*P(n,x) = (n^3-2*n^2+n+4+(4*n^3-24*n^2+49*n-55)*x+(16*n^2-72*n+92)*x^2+(16*n-32)*x^3)*P(n-1,x) + ((2*n^2-6*n+6)*x-(8*n^2-32*n+38)*x^2-(32*n-64)*x^3-32*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 - 2*n^2 + n + 4 + (4*n^3 - 24*n^2 + 49*n - 55)*x + (16*n^2 - 72*n + 92)*x^2 + (16*n - 32)*x^3)*P[n - 1, x] + ((2*n^2 - 6*n + 6)*x - (8*n^2 - 32*n + 38)*x^2 - (32*n - 64)*x^3 - 32*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 3*n + 8 + (4*n^2 - 11*n - 15)*x + (4*n + 4)*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 - 2*n^2 + n + 4 + (4*n^3 - 24*n^2 + 49*n - 55)*x + (16*n^2 - 72*n + 92)*x^2 + (16*n - 32)*x^3)*v[i - 1] + ((2*n^2 - 6*n + 6)*x - (8*n^2 - 32*n + 38)*x^2 - (32*n - 64)*x^3 - 32*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 3*n + 8 + (4*n^2 - 11*n - 15)*x + (4*n + 4)*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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023