Real-rooted atlas

A052553 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-30 - 18\,x + 30\,n - 27\,n x + 27\,n x^{2} + 45\,n^{2} x - 54\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-150 - 540\,x - 432\,x^{2} + 90\,n - 36\,n x - 729\,n x^{2} - 324\,n x^{3} + 180\,n^{2} x + 324\,n^{2} x^{2} - 162\,n^{2} x^{3} + 81\,n^{3} x^{2} + 162\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 165\,x - 342\,x^{2} - 216\,x^{3} - 75\,n x - 450\,n x^{2} - 783\,n x^{3} - 324\,n x^{4} - 45\,n^{2} x^{2} - 243\,n^{2} x^{3} - 324\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3268
431215
55302762
686081149
713120243108484
8212285944321275
934426137715124053754
105578029974320202510177
118914106291113408100145828688
121442520127982764826325874878183
132334464254346415276950393665103215702
1437778484957214299220695514288435721586349
156101371095094308880524475459270178605174961598140
Technical details
Node id
n-045e1c27e91fdcc6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8aaabef611692c95

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 -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:  1;
   3:  2,   6;
   4:  3,  12;
   5:  5,  30,   27;
   6:  8,  60,   81;
   7: 13, 120,  243,  108;
   8: 21, 228,  594,  432;
   9: 34, 426, 1377, 1512,  405;
  10: 55, 780, 2997, 4320, 2025;
%F Formula
The row polynomials satisfy (30*n-30+(45*n^2-27*n-18)*x+(27*n^3-54*n^2+27*n)*x^2)*P(n,x) = (90*n-150+(180*n^2-36*n-540)*x+(81*n^3+324*n^2-729*n-432)*x^2+(162*n^3-162*n^2-324*n)*x^3)*P(n-1,x) - (25+(75*n+165)*x+(45*n^2+450*n+342)*x^2+(243*n^2+783*n+216)*x^3+(324*n^2+324*n)*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[((90*n - 150 + (180*n^2 - 36*n - 540)*x + (81*n^3 + 324*n^2 - 729*n - 432)*x^2 + (162*n^3 - 162*n^2 - 324*n)*x^3)*P[n - 1, x] - (25 + (75*n + 165)*x + (45*n^2 + 450*n + 342)*x^2 + (243*n^2 + 783*n + 216)*x^3 + (324*n^2 + 324*n)*x^4)*D[P[n - 1, x], x])/(30*n - 30 + (45*n^2 - 27*n - 18)*x + (27*n^3 - 54*n^2 + 27*n)*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] = ((90*n - 150 + (180*n^2 - 36*n - 540)*x + (81*n^3 + 324*n^2 - 729*n - 432)*x^2 + (162*n^3 - 162*n^2 - 324*n)*x^3)*v[i - 1] - (25 + (75*n + 165)*x + (45*n^2 + 450*n + 342)*x^2 + (243*n^2 + 783*n + 216)*x^3 + (324*n^2 + 324*n)*x^4)*deriv(v[i - 1], x))/(30*n - 30 + (45*n^2 - 27*n - 18)*x + (27*n^3 - 54*n^2 + 27*n)*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. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.