Real-rooted atlas

narayana → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31620541
4641448414306
5256896896336422426
61024512076804800132013220076
74096276485760052800237605148429171481
8163841433603942404928003203201121122002014301500666

All 15 rows

Technical details
Node id
n-990f00bddc02dc6e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c585b23acdf03520

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:      4,       2;
   3:     16,      20,        5;
   4:     64,     144,       84,       14;
   5:    256,     896,      896,      336,       42;
   6:   1024,    5120,     7680,     4800,     1320,      132;
   7:   4096,   27648,    57600,    52800,    23760,     5148,     429;
   8:  16384,  143360,   394240,   492800,   320320,   112112,   20020,    1430;
   9:  65536,  720896,  2523136,  4100096,  3587584,  1793792,  512512,   77792,   4862;
  10: 262144, 3538944, 15335424, 31309824, 35223552, 23482368, 9504768, 2291328, 302328, 16796;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (4*n+4+(2*n+2)*x)*P(n-1,x) + (8*x+2*x^2)*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[((4*n + 4 + (2*n + 2)*x)*P[n - 1, x] + (8*x + 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
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] = ((4*n + 4 + (2*n + 2)*x)*v[i - 1] + (8*x + 2*x^2)*deriv(v[i - 1], x))/(n + 1)); 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