Real-rooted atlas

narayana → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3510318
4144227487
542168168565439
613266090048010062278
74292574445533001125162712052
8143010010210212002096252310245864669

All 15 rows

Technical details
Node id
n-3bda2fd99a20605f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1dcc13434ed0f3e9

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     2,      2;
   3:     5,     10,      3;
   4:    14,     42,     27,      4;
   5:    42,    168,    168,     56,      5;
   6:   132,    660,    900,    480,    100,      6;
   7:   429,   2574,   4455,   3300,   1125,    162,     7;
   8:  1430,  10010,  21021,  20020,   9625,   2310,   245,    8;
   9:  4862,  38896,  96096, 112112,  70070,  24024,  4312,  352,   9;
  10: 16796, 151164, 429624, 594048, 458640, 206388, 53508, 7488, 486, 10;
%F Formula
The row polynomials satisfy (2*n^2+2*n+(n^3-n)*x)*P(n,x) = (8*n^2-4*n+(4*n^3+6*n^2-10*n)*x+(3*n^3-2*n^2-3*n+2)*x^2)*P(n-1,x) - (4*n*x+(2*n^2+5*n-1)*x^2+(2*n^2+n-1)*x^3)*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[((8*n^2 - 4*n + (4*n^3 + 6*n^2 - 10*n)*x + (3*n^3 - 2*n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (4*n*x + (2*n^2 + 5*n - 1)*x^2 + (2*n^2 + n - 1)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 2*n + (n^3 - n)*x)]];
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] = ((8*n^2 - 4*n + (4*n^3 + 6*n^2 - 10*n)*x + (3*n^3 - 2*n^2 - 3*n + 2)*x^2)*v[i - 1] - (4*n*x + (2*n^2 + 5*n - 1)*x^2 + (2*n^2 + n - 1)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 2*n + (n^3 - n)*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