Real-rooted atlas

A008459 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-8 - 6\,x + 13\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-8 - 14\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
23336978
324538718927848
4192139842916864819766
5155253991539060178203645243116208
612790539258048154529808095985145807291411404
71067925381055556218054402755194764547457956133218717383584

All 15 rows

Other recurrences

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

\(\left(n\right)\,P_{n}(x) = \left(9 + 18\,x + 9\,x^{2} - 9\,n - 18\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-5 - 3\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)
Technical details
Node id
n-2a830d2c1dad85e9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-498b697045e9102b)
Fingerprint
f6-54d1a54334315095

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        5,         3;
  2:       33,        36,         9;
  3:      245,       387,       189,        27;
  4:     1921,      3984,      2916,       864,        81;
  5:    15525,     39915,     39060,     17820,      3645,       243;
  6:   127905,    392580,    481545,    298080,     95985,     14580,      729;
  7:  1067925,   3810555,   5621805,   4402755,   1947645,    474579,    56133,    2187;
  8:  9004545,  36624960,  63156240,  59875200,  33781860,  11430720,  2204496,  209952,   6561;
  9: 76499525, 349356915, 689530320, 767718000, 527344020, 229206348, 61970832, 9762768, 767637, 19683;
%F Formula
The row polynomials satisfy n*P(n,x) = (13*n-8+(9*n-6)*x)*P(n-1,x) - (8+14*x+6*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((13*n - 8 + (9*n - 6)*x)*P[n - 1, x] - (8 + 14*x + 6*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((13*n - 8 + (9*n - 6)*x)*v[i - 1] - (8 + 14*x + 6*x^2)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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