Real-rooted atlas

A026729 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-30 - 72\,x + 30\,n + 27\,n x - 54\,n x^{2} + 45\,n^{2} x + 27\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(150 + 360\,x + 162\,x^{2} - 60\,n + 261\,n x + 675\,n x^{2} + 324\,n x^{3} - 135\,n^{2} x + 27\,n^{2} x^{2} + 162\,n^{2} x^{3} - 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 135\,x + 207\,x^{2} + 81\,x^{3} + 75\,n x + 360\,n x^{2} + 459\,n x^{3} + 162\,n x^{4} + 45\,n^{2} x^{2} + 162\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2167
32182747
4342135108288
55904327564051688
6818011613132364514589584
7133482808102601863016038510353200
82165463452926872495976866633917496290304

All 15 rows

Technical details
Node id
n-21eb87379f43dd61 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a636c51e6eaf8091

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:  1,    6;
   3:  2,   18,    27;
   4:  3,   42,   135,    108;
   5:  5,   90,   432,    756,    405;
   6:  8,  180,  1161,   3132,   3645,    1458;
   7: 13,  348,  2808,  10260,  18630,   16038,    5103;
   8: 21,  654,  6345,  29268,  72495,   97686,   66339,   17496;
   9: 34, 1206, 13662,  76140, 239355,  441774,  469476,  262440,   59049;
  10: 55, 2190, 28377, 185436, 707130, 1662120, 2423925, 2117016, 1003833, 196830;
%F Formula
The row polynomials satisfy (30*n-30+(45*n^2+27*n-72)*x+(27*n^3+27*n^2-54*n)*x^2)*P(n,x) = -(60*n-150+(135*n^2-261*n-360)*x+(54*n^3-27*n^2-675*n-162)*x^2-(162*n^2+324*n)*x^3)*P(n-1,x) + (25+(75*n+135)*x+(45*n^2+360*n+207)*x^2+(162*n^2+459*n+81)*x^3+(81*n^2+162*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[(-(60*n - 150 + (135*n^2 - 261*n - 360)*x + (54*n^3 - 27*n^2 - 675*n - 162)*x^2 - (162*n^2 + 324*n)*x^3)*P[n - 1, x] + (25 + (75*n + 135)*x + (45*n^2 + 360*n + 207)*x^2 + (162*n^2 + 459*n + 81)*x^3 + (81*n^2 + 162*n)*x^4)*D[P[n - 1, x], x])/(30*n - 30 + (45*n^2 + 27*n - 72)*x + (27*n^3 + 27*n^2 - 54*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] = (-(60*n - 150 + (135*n^2 - 261*n - 360)*x + (54*n^3 - 27*n^2 - 675*n - 162)*x^2 - (162*n^2 + 324*n)*x^3)*v[i - 1] + (25 + (75*n + 135)*x + (45*n^2 + 360*n + 207)*x^2 + (162*n^2 + 459*n + 81)*x^3 + (81*n^2 + 162*n)*x^4)*deriv(v[i - 1], x))/(30*n - 30 + (45*n^2 + 27*n - 72)*x + (27*n^3 + 27*n^2 - 54*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

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.