Real-rooted atlas

A055151 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 12\,x + 6\,n + 14\,n x + 2\,n x^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-12 - 48\,x - 48\,x^{2} - 8\,x^{3} + 30\,n + 52\,n x + 14\,n x^{2} - 4\,n x^{3} + 26\,n^{2} x + 35\,n^{2} x^{2} + 8\,n^{2} x^{3} + 5\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-18 - 78\,x - 120\,x^{2} - 76\,x^{3} - 16\,x^{4} - 24\,n x - 74\,n x^{2} - 74\,n x^{3} - 24\,n x^{4} - 6\,n^{2} x^{2} - 14\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2224
34610
4920635
5216030111
65118013520386
71275325251401324

All 15 rows

Technical details
Node id
n-4afc9bb245d764ac · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-169fac48b5c9ac7c

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   1;
  2:   2,    2;
  3:   4,    6;
  4:   9,   20,    6;
  5:  21,   60,   30;
  6:  51,  180,  135,   20;
  7: 127,  532,  525,  140;
  8: 323, 1568, 1932,  784,  70;
  9: 835, 4608, 6804, 3696, 630;
%F Formula
The row polynomials satisfy (6*n+12+(4*n^2+14*n+12)*x+(n^3+3*n^2+2*n)*x^2)*P(n,x) = (30*n-12+(26*n^2+52*n-48)*x+(5*n^3+35*n^2+14*n-48)*x^2+(4*n^3+8*n^2-4*n-8)*x^3)*P(n-1,x) - (18+(24*n+78)*x+(6*n^2+74*n+120)*x^2+(14*n^2+74*n+76)*x^3+(8*n^2+24*n+16)*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[((30*n - 12 + (26*n^2 + 52*n - 48)*x + (5*n^3 + 35*n^2 + 14*n - 48)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*P[n - 1, x] - (18 + (24*n + 78)*x + (6*n^2 + 74*n + 120)*x^2 + (14*n^2 + 74*n + 76)*x^3 + (8*n^2 + 24*n + 16)*x^4)*D[P[n - 1, x], x])/(6*n + 12 + (4*n^2 + 14*n + 12)*x + (n^3 + 3*n^2 + 2*n)*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] = ((30*n - 12 + (26*n^2 + 52*n - 48)*x + (5*n^3 + 35*n^2 + 14*n - 48)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*v[i - 1] - (18 + (24*n + 78)*x + (6*n^2 + 74*n + 120)*x^2 + (14*n^2 + 74*n + 76)*x^3 + (8*n^2 + 24*n + 16)*x^4)*deriv(v[i - 1], x))/(6*n + 12 + (4*n^2 + 14*n + 12)*x + (n^3 + 3*n^2 + 2*n)*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

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