Real-rooted atlas

A053117 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 5\,x - x^{2} + n - 15\,n x + 5\,n x^{2} + 15\,n^{2} x - 10\,n^{2} x^{2} - 5\,n^{3} x + 10\,n^{3} x^{2} - 5\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(n - n x - n x^{2} + 10\,n^{2} x + 5\,n^{2} x^{2} - 5\,n^{3} x - 3\,n^{3} x^{2} + n^{3} x^{3} - 4\,n^{4} x^{2} - 2\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - x^{2} - x^{3} + 10\,n x^{2} + 5\,n x^{3} - 10\,n^{2} x^{2} - 3\,n^{2} x^{3} + n^{2} x^{4} - 4\,n^{3} x^{3} - 2\,n^{3} x^{4} + 2\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
22810
33323671
4480216128428
5516075610244002345
66280201646084000115212062
774484536153602200013824313659311
886729072422408800089856439048192281944

All 15 rows

Technical details
Node id
n-c8a1244763a9c604 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b38c31b44d930868

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:  2,    8;
   3:  3,   32,    36;
   4:  4,   80,   216,    128;
   5:  5,  160,   756,   1024,    400;
   6:  6,  280,  2016,   4608,   4000,    1152;
   7:  7,  448,  4536,  15360,  22000,   13824,    3136;
   8:  8,  672,  9072,  42240,  88000,   89856,   43904,    8192;
   9:  9,  960, 16632, 101376, 286000,  419328,  329280,  131072,  20736;
  10: 10, 1320, 28512, 219648, 800800, 1572480, 1756160, 1114112, 373248, 51200;
%F Formula
The row polynomials satisfy (n-1-(5*n^3-15*n^2+15*n-5)*x+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^2)*P(n,x) = (n-(5*n^3-10*n^2+n)*x+(n^5-4*n^4-3*n^3+5*n^2-n)*x^2+(n^5-2*n^4+n^3)*x^3)*P(n-1,x) + (2*x-(10*n^2-10*n+1)*x^2+(2*n^4-4*n^3-3*n^2+5*n-1)*x^3+(n^4-2*n^3+n^2)*x^4)*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[((n - (5*n^3 - 10*n^2 + n)*x + (n^5 - 4*n^4 - 3*n^3 + 5*n^2 - n)*x^2 + (n^5 - 2*n^4 + n^3)*x^3)*P[n - 1, x] + (2*x - (10*n^2 - 10*n + 1)*x^2 + (2*n^4 - 4*n^3 - 3*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (5*n^3 - 15*n^2 + 15*n - 5)*x + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^2)]];
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] = ((n - (5*n^3 - 10*n^2 + n)*x + (n^5 - 4*n^4 - 3*n^3 + 5*n^2 - n)*x^2 + (n^5 - 2*n^4 + n^3)*x^3)*v[i - 1] + (2*x - (10*n^2 - 10*n + 1)*x^2 + (2*n^4 - 4*n^3 - 3*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (5*n^3 - 15*n^2 + 15*n - 5)*x + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023