Real-rooted atlas

narayana → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-11 - 9\,x + 13\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-8 - 14\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
32933971
418530616227680
5125727182124702816882
68925238052457012150283524372528
7654452074952660851753656196510935729788019
849182518064202768220228123010829702925724082421878766248

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 + n\right)\,P_{n}(x) = \left(18 + 36\,x + 18\,x^{2} - 9\,n - 18\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-5 - 3\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)
Technical details
Node id
n-04d8e8670fe3c4ee · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-aa2e4d5fc01bc97a)
Fingerprint
f6-b59858f0c3c98e59

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:        5,         3;
   3:       29,        33,         9;
   4:      185,       306,       162,        27;
   5:     1257,      2718,      2124,       702,        81;
   6:     8925,     23805,     24570,     12150,      2835,       243;
   7:    65445,    207495,    266085,    175365,     61965,     10935,      729;
   8:   491825,   1806420,   2768220,   2281230,   1082970,    292572,    40824,    2187;
   9:  3768209,  15731052,  28039536,  27743688,  16563204,   6055560,  1306368,  148716,   6561;
  10: 29324405, 137122983, 278688924, 321906312, 231737436, 107101764, 31475304, 5589972, 531441, 19683;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (13*n-11+(9*n-9)*x)*P(n-1,x) - (8+14*x+6*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[((13*n - 11 + (9*n - 9)*x)*P[n - 1, x] - (8 + 14*x + 6*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] = ((13*n - 11 + (9*n - 9)*x)*v[i - 1] - (8 + 14*x + 6*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