Real-rooted atlas

narayana → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - n + 4\,n x + n^{2} + 3\,n^{2} x - 4\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(4\,x + 2\,x^{2} + n - 12\,n x - 5\,n x^{2} + n^{2} + 5\,n^{2} x + 14\,n^{2} x^{2} - 2\,n^{3} x - 12\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x + x^{3} + 2\,n x - 2\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3347
4624939
510809016196
61520045024025930
72142015751680525364257
82878444107840490010084919019
93613441058428224294001209617646483512
1045216022680846721323009072026460288081361998
11553300445502217604851004989602425505280044551001553630
12664840816755227201524600219542416008305808009801066001216615686
13786864141570113256042471008154432832431645302401274130171600943814427992472
1491946423423422902881073572526501904360720362748345611594583260260028628613104169117823940
1510512740372645437236825050025772972201352701351374172808116208127327300501000545864017745196493768485
Technical details
Node id
n-6724d1657fe35b01 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b29d29aac1e7588a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:  0;
   2:  1;
   3:  3,    4;
   4:  6,   24,     9;
   5: 10,   80,    90,    16;
   6: 15,  200,   450,   240,     25;
   7: 21,  420,  1575,  1680,    525,    36;
   8: 28,  784,  4410,  7840,   4900,  1008,    49;
   9: 36, 1344, 10584, 28224,  29400, 12096,  1764,   64;
  10: 45, 2160, 22680, 84672, 132300, 90720, 26460, 2880, 81;
%F Formula
The row polynomials satisfy (n^2-n-2+(n^4-4*n^3+3*n^2+4*n-4)*x)*P(n,x) = (n^2+n+(n^4-2*n^3+5*n^2-12*n+4)*x+(3*n^4-12*n^3+14*n^2-5*n+2)*x^2)*P(n-1,x) + ((2*n-1)*x+(2*n^3-4*n^2)*x^2-(2*n^3-4*n^2+2*n-1)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (n^4 - 2*n^3 + 5*n^2 - 12*n + 4)*x + (3*n^4 - 12*n^3 + 14*n^2 - 5*n + 2)*x^2)*P[n - 1, x] + ((2*n - 1)*x + (2*n^3 - 4*n^2)*x^2 - (2*n^3 - 4*n^2 + 2*n - 1)*x^3)*D[P[n - 1, x], x])/(n^2 - n - 2 + (n^4 - 4*n^3 + 3*n^2 + 4*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 + n + (n^4 - 2*n^3 + 5*n^2 - 12*n + 4)*x + (3*n^4 - 12*n^3 + 14*n^2 - 5*n + 2)*x^2)*v[i - 1] + ((2*n - 1)*x + (2*n^3 - 4*n^2)*x^2 - (2*n^3 - 4*n^2 + 2*n - 1)*x^3)*deriv(v[i - 1], x))/(n^2 - n - 2 + (n^4 - 4*n^3 + 3*n^2 + 4*n - 4)*x)); 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