Real-rooted atlas

A008459 → x ↦ x+1 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 2\,x + 7\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2138122
36357151136
4321384156241886
51683250513803403515944
689891600811145380064548140636
748639100849849033706588551113631281488

All 15 rows

Other recurrences

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

\(\left(n\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-3 - x + 6\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-c8c2604dc29a0657 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-577981c2154866df)
Fingerprint
f6-c880512e8efe320e

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:       3,       1;
  2:      13,       8,       1;
  3:      63,      57,      15,       1;
  4:     321,     384,     156,      24,       1;
  5:    1683,    2505,    1380,     340,      35,      1;
  6:    8989,   16008,   11145,    3800,     645,     48,     1;
  7:   48639,  100849,   84903,   37065,    8855,   1113,    63,    1;
  8:  265729,  628736,  621040,  330848,  102340,  18368,  1792,   80,  1;
  9: 1462563, 3888657, 4408848, 2774352, 1055628, 247716, 34944, 2736, 99, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (7*n-4+(3*n-2)*x)*P(n-1,x) - (4+6*x+2*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[((7*n - 4 + (3*n - 2)*x)*P[n - 1, x] - (4 + 6*x + 2*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] = ((7*n - 4 + (3*n - 2)*x)*v[i - 1] - (4 + 6*x + 2*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

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