Real-rooted atlas

narayana → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-8 - 6\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31918441
4100138608306
55621016656176162426
633047380632025604803220076
7200715335856940309208880124864171481
81249963853224924363357201304802844831361281500666

All 15 rows

Other recurrences

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

\(\left(1 + n\right)\,P_{n}(x) = \left(8 + 16\,x + 8\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)
Technical details
Node id
n-d7a98327de1b7eb5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-9731fbb885e8c59b)
Fingerprint
f6-1afe30f8794f3fd0

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:      19,       18,        4;
   4:     100,      138,       60,        8;
   5:     562,     1016,      656,      176,       16;
   6:    3304,     7380,     6320,     2560,      480,      32;
   7:   20071,    53358,    56940,    30920,     8880,    1248,      64;
   8:  124996,   385322,   492436,   335720,   130480,   28448,    3136,    128;
   9:  793774,  2783376,  4144224,  3402336,  1671264,  497280,   86016,   7680,   256;
  10: 5120632, 20124540, 34210080, 32860800, 19535040, 7390656, 1757952, 248832, 18432, 512;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (10*n-8+(6*n-6)*x)*P(n-1,x) - (6+10*x+4*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[((10*n - 8 + (6*n - 6)*x)*P[n - 1, x] - (6 + 10*x + 4*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] = ((10*n - 8 + (6*n - 6)*x)*v[i - 1] - (6 + 10*x + 4*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