Real-rooted atlas

A001263 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-10\,x + 6\,x^{2} - 2\,n + 7\,n x + 3\,n x^{2} + 5\,n^{2} + 6\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(-10\,x + 6\,x^{2} + 4\,n - 17\,n x - 6\,n x^{2} + 27\,n x^{3} + 14\,n^{2} + 33\,n^{2} x + 54\,n^{2} x^{2} + 12\,n^{3} + 27\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x - 8\,x^{2} - 12\,x^{3} - 18\,x^{4} - 7\,n x - 33\,n x^{2} - 36\,n x^{3} - 6\,n^{2} x - 18\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
22033962
37722516227491
429412601638702813975
511226390124209720283524332730
642903059180190951755103010935729272940
7164451411414684687588356237002466454082421872298245

All 15 rows

Technical details
Node id
n-d072492d6b036063 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f603151ffaefdb02

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:     20,      33,        9;
  3:     77,     225,      162,       27;
  4:    294,    1260,     1638,      702,       81;
  5:   1122,    6390,    12420,     9720,     2835,      243;
  6:   4290,   30591,    80190,    95175,    51030,    10935,      729;
  7:  16445,  141141,   468468,   758835,   623700,   246645,    40824,    2187;
  8:  63206,  634920,  2558556,  5297292,  5999994,  3667356,  1122660,  148716,   6561;
  9: 243542, 2804490, 13318344, 33729696, 49223538, 41793570, 19901700, 4881384, 531441, 19683;
%F Formula
The row polynomials satisfy (3*n^3+5*n^2-2*n+(6*n^2+7*n-10)*x+(3*n+6)*x^2)*P(n,x) = (12*n^3+14*n^2+4*n+(27*n^3+33*n^2-17*n-10)*x+(54*n^2-6*n+6)*x^2+27*n*x^3)*P(n-1,x) - ((6*n^2+7*n+2)*x+(18*n^2+33*n+8)*x^2+(36*n+12)*x^3+18*x^4)*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[((12*n^3 + 14*n^2 + 4*n + (27*n^3 + 33*n^2 - 17*n - 10)*x + (54*n^2 - 6*n + 6)*x^2 + 27*n*x^3)*P[n - 1, x] - ((6*n^2 + 7*n + 2)*x + (18*n^2 + 33*n + 8)*x^2 + (36*n + 12)*x^3 + 18*x^4)*D[P[n - 1, x], x])/(3*n^3 + 5*n^2 - 2*n + (6*n^2 + 7*n - 10)*x + (3*n + 6)*x^2)]];
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] = ((12*n^3 + 14*n^2 + 4*n + (27*n^3 + 33*n^2 - 17*n - 10)*x + (54*n^2 - 6*n + 6)*x^2 + 27*n*x^3)*v[i - 1] - ((6*n^2 + 7*n + 2)*x + (18*n^2 + 33*n + 8)*x^2 + (36*n + 12)*x^3 + 18*x^4)*deriv(v[i - 1], x))/(3*n^3 + 5*n^2 - 2*n + (6*n^2 + 7*n - 10)*x + (3*n + 6)*x^2)); 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