Real-rooted atlas

narayana → basis A013610 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(18 + 108\,x + 162\,x^{2} - 9\,n - 54\,n x - 81\,n x^{2}\right)\,P_{n-2}(x) + \left(-5 - 9\,x + 10\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
25914
3299981209
418591814587293290
5125781541911618954656154042
689257141522113032805022963559049918204
765445622485239476547348555019165265720553144116025361
8491825541926024913980615932108772057071094996297606964782969285777506

All 15 rows

Technical details
Node id
n-8a459424c405b599 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dceb48071bfd4833

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,         9;
   3:       29,        99,         81;
   4:      185,       918,       1458,        729;
   5:     1257,      8154,      19116,      18954,        6561;
   6:     8925,     71415,     221130,     328050,      229635,       59049;
   7:    65445,    622485,    2394765,    4734855,     5019165,     2657205,      531441;
   8:   491825,   5419260,   24913980,   61593210,    87720570,    71094996,    29760696,     4782969;
   9:  3768209,  47193156,  252355824,  749079576,  1341619524,  1471501080,   952342272,   325241892,   43046721;
  10: 29324405, 411368949, 2508200316, 8691470424, 18770732316, 26025728652, 22945496616, 12225268764, 3486784401, 387420489;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(9*n-18+(54*n-108)*x+(81*n-162)*x^2)*P(n-2,x) + (10*n-5+(18*n-9)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 9*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 9*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n - 18 + (54*n - 108)*x + (81*n - 162)*x^2)*P[n - 2, x] + (10*n - 5 + (18*n - 9)*x)*P[n - 1, 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; v[2] = 5 + 9*x; for(n = 3, N, my(i = n); v[i] = (-(9*n - 18 + (54*n - 108)*x + (81*n - 162)*x^2)*v[i - 2] + (10*n - 5 + (18*n - 9)*x)*v[i - 1])/(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