Real-rooted atlas

A125662 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + 6\,n^{2} x - 10\,n^{2} x^{2} + 12\,n^{3} x + 10\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-6\,x - 24\,x^{2} - 30\,x^{3} + 2\,n + 6\,n x - 33\,n x^{2} - 90\,n x^{3} - 25\,n x^{4} + 4\,n^{2} + 42\,n^{2} x + 101\,n^{2} x^{2} + 45\,n^{2} x^{3} + 12\,n^{3} x + 64\,n^{3} x^{2} + 75\,n^{3} x^{3} + 25\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 9\,x^{2} - 23\,x^{3} - 15\,x^{4} - 2\,n x - 24\,n x^{2} - 87\,n x^{3} - 90\,n x^{4} - 25\,n x^{5} - 6\,n^{2} x^{2} - 41\,n^{2} x^{3} - 60\,n^{2} x^{4} - 25\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
21122437
31187584178
4124153360275813
513025893615008643589
6136390192047555724263915465
71425493420115252143820573789665444

All 15 rows

Technical details
Node id
n-6477786aeb94a670 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-660f1d3576ddd75e

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: 1,  6;
  2: 1, 12,  24;
  3: 1, 18,  75,   84;
  4: 1, 24, 153,  360,   275;
  5: 1, 30, 258,  936,  1500,    864;
  6: 1, 36, 390, 1920,  4755,   5724,   2639;
  7: 1, 42, 549, 3420, 11525,  21438,  20573,   7896;
  8: 1, 48, 735, 5544, 23700,  60048,  88949,  70800,  23256;
  9: 1, 54, 948, 8400, 43575, 140238, 282982, 347184, 235755, 67650;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(12*n^3+6*n^2)*x+(10*n^3-10*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(12*n^3+42*n^2+6*n-6)*x+(64*n^3+101*n^2-33*n-24)*x^2+(75*n^3+45*n^2-90*n-30)*x^3+(25*n^3-25*n)*x^4)*P(n-1,x) - ((2*n+1)*x+(6*n^2+24*n+9)*x^2+(41*n^2+87*n+23)*x^3+(60*n^2+90*n+15)*x^4+(25*n^2+25*n)*x^5)*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[((4*n^2 + 2*n + (12*n^3 + 42*n^2 + 6*n - 6)*x + (64*n^3 + 101*n^2 - 33*n - 24)*x^2 + (75*n^3 + 45*n^2 - 90*n - 30)*x^3 + (25*n^3 - 25*n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (41*n^2 + 87*n + 23)*x^3 + (60*n^2 + 90*n + 15)*x^4 + (25*n^2 + 25*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (10*n^3 - 10*n^2)*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] = ((4*n^2 + 2*n + (12*n^3 + 42*n^2 + 6*n - 6)*x + (64*n^3 + 101*n^2 - 33*n - 24)*x^2 + (75*n^3 + 45*n^2 - 90*n - 30)*x^3 + (25*n^3 - 25*n)*x^4)*v[i - 1] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (41*n^2 + 87*n + 23)*x^3 + (60*n^2 + 90*n + 15)*x^4 + (25*n^2 + 25*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (10*n^3 - 10*n^2)*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. [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