Real-rooted atlas

A110441 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + 6\,n^{2} x - 2\,n^{2} x^{2} + 12\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-6\,x - 12\,x^{2} - 6\,x^{3} + 2\,n + 6\,n x - 21\,n x^{2} - 18\,n x^{3} - n x^{4} + 4\,n^{2} + 42\,n^{2} x + 85\,n^{2} x^{2} + 9\,n^{2} x^{3} + 12\,n^{3} x + 56\,n^{3} x^{2} + 15\,n^{3} x^{3} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 9\,x^{2} - 19\,x^{3} - 3\,x^{4} - 2\,n x - 24\,n x^{2} - 75\,n x^{3} - 18\,n x^{4} - n x^{5} - 6\,n^{2} x^{2} - 37\,n^{2} x^{3} - 12\,n^{2} x^{4} - n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
21122134
31186960148
4124144288155612
513024679210053782452
61363751680349531328899608
714253130608975133389037204037024

All 15 rows

Technical details
Node id
n-275d0e075f661270 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-61571462e09ff891

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,  21;
  3: 1, 18,  69,   60;
  4: 1, 24, 144,  288,   155;
  5: 1, 30, 246,  792,  1005,    378;
  6: 1, 36, 375, 1680,  3495,   3132,    889;
  7: 1, 42, 531, 3060,  8975,  13338,   9037,   2040;
  8: 1, 48, 714, 5040, 19200,  40608,  45934,  24672,  4599;
  9: 1, 54, 924, 7728, 36330, 100548, 162932, 146592, 64629, 10230;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(12*n^3+6*n^2)*x+(2*n^3-2*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(12*n^3+42*n^2+6*n-6)*x+(56*n^3+85*n^2-21*n-12)*x^2+(15*n^3+9*n^2-18*n-6)*x^3+(n^3-n)*x^4)*P(n-1,x) - ((2*n+1)*x+(6*n^2+24*n+9)*x^2+(37*n^2+75*n+19)*x^3+(12*n^2+18*n+3)*x^4+(n^2+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 + (56*n^3 + 85*n^2 - 21*n - 12)*x^2 + (15*n^3 + 9*n^2 - 18*n - 6)*x^3 + (n^3 - n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (37*n^2 + 75*n + 19)*x^3 + (12*n^2 + 18*n + 3)*x^4 + (n^2 + n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (2*n^3 - 2*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 + (56*n^3 + 85*n^2 - 21*n - 12)*x^2 + (15*n^3 + 9*n^2 - 18*n - 6)*x^3 + (n^3 - n)*x^4)*v[i - 1] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (37*n^2 + 75*n + 19)*x^3 + (12*n^2 + 18*n + 3)*x^4 + (n^2 + n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (2*n^3 - 2*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. [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