Real-rooted atlas

narayana → basis A007318 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x - 6\,n - 7\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3527
42118342
584112424242
63306003608051375
712872970247590013567773
85005140141501577001925210743876
9194486406484084560562002036963088247684
107558228641644553636691217199045864655243291399293
1129393012597002267460222768012994804586409555010920585107913955
12114406654716201119195012790800895356039584161099560184800173257701144812878
13445740023535072539345407034940057558600306979201074427224235203366002640099012254064726
141738386010044008025496328037394614435057451021949012893117024266048644988412583440388961248131442131899
15678639154259045701186448445193347154020450179751472412942736206471256071816611080479699690969969556921547148195232633
Technical details
Node id
n-a9b81a684a479992 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-63d66b489de6a076

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:     5,      2;
   4:    21,     18,      3;
   5:    84,    112,     42,      4;
   6:   330,    600,    360,     80,      5;
   7:  1287,   2970,   2475,    900,    135,     6;
   8:  5005,  14014,  15015,   7700,   1925,   210,    7;
   9: 19448,  64064,  84084,  56056,  20020,  3696,  308,   8;
  10: 75582, 286416, 445536, 366912, 171990, 45864, 6552, 432, 9;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = (4*n^2-6*n+2+(3*n^2-7*n+2)*x)*P(n-1,x) - ((2*n-1)*x+(2*n-1)*x^2)*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[((4*n^2 - 6*n + 2 + (3*n^2 - 7*n + 2)*x)*P[n - 1, x] - ((2*n - 1)*x + (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n - 2)]];
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] = ((4*n^2 - 6*n + 2 + (3*n^2 - 7*n + 2)*x)*v[i - 1] - ((2*n - 1)*x + (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n - 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