Real-rooted atlas

narayana → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(2 + 4\,x - 6\,n - 14\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} - 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
310818
4427224138
5168448336641016
66602400288012801607380
72574118801980014400432038453358
810010560561201201232006160013440896385322

All 15 rows

Technical details
Node id
n-25810b399b931dcd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3ac782335c346aee

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:      2;
   3:     10,       8;
   4:     42,      72,      24;
   5:    168,     448,     336,      64;
   6:    660,    2400,    2880,    1280,     160;
   7:   2574,   11880,   19800,   14400,    4320,     384;
   8:  10010,   56056,  120120,  123200,   61600,   13440,    896;
   9:  38896,  256256,  672672,  896896,  640640,  236544,  39424,   2048;
  10: 151164, 1145664, 3564288, 5870592, 5503680, 2935296, 838656, 110592, 4608;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = (4*n^2-6*n+2+(6*n^2-14*n+4)*x)*P(n-1,x) - ((2*n-1)*x+(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (6*n^2 - 14*n + 4)*x)*P[n - 1, x] - ((2*n - 1)*x + (4*n - 2)*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] = 2; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 6*n + 2 + (6*n^2 - 14*n + 4)*x)*v[i - 1] - ((2*n - 1)*x + (4*n - 2)*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