Real-rooted atlas

A008292 → γ (shift +1) → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3\,x + 2\,n x^{3}\right)\,P_{n}(x) = \left(-54 - 144\,x - 60\,x^{2} + 96\,x^{3} + 64\,x^{4} + 36\,n + 60\,n x + 12\,n x^{2} + 32\,n x^{3} + 64\,n x^{4} + 16\,n x^{5} + 12\,n^{2} x - 36\,n^{2} x^{3} - 16\,n^{2} x^{4} + 8\,n^{2} x^{5} - 8\,n^{3} x^{3} - 16\,n^{3} x^{4} - 8\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-27 - 99\,x - 102\,x^{2} + 18\,x^{3} + 80\,x^{4} + 32\,x^{5} - 18\,n x - 42\,n x^{2} + 12\,n x^{3} + 96\,n x^{4} + 76\,n x^{5} + 16\,n x^{6} + 12\,n^{2} x^{3} + 40\,n^{2} x^{4} + 44\,n^{2} x^{5} + 16\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-18 - 9\,x + 24\,x^{2} + 8\,x^{3} - 12\,n x + 10\,n x^{3} - 8\,n x^{4} + 8\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(9\,x + 21\,x^{2} + 12\,x^{3} - 6\,n x^{3} - 14\,n x^{4} - 8\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2347
391625
43910848195
51896484081245
6110747404608108811543
77281365764492815872104657

All 15 rows

Technical details
Node id
n-abdfd8edac4c2935 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5e04371f1947d221

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 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;
  2:      3,       4;
  3:      9,      16;
  4:     39,     108,      48;
  5:    189,     648,     408;
  6:   1107,    4740,    4608,    1088;
  7:   7281,   36576,   44928,   15872;
  8:  54351,  316764,  486432,  264192,  39680;
  9: 448821, 2961144, 5377752, 3750912, 884480;
%F Formula
The row polynomials satisfy -(3*x-2*n*x^3)*P(n,x) = (36*n-54+(12*n^2+60*n-144)*x+(12*n-60)*x^2-(8*n^3+36*n^2-32*n-96)*x^3-(16*n^3+16*n^2-64*n-64)*x^4-(8*n^3-8*n^2-16*n)*x^5)*P(n-2,x) - (27+(18*n+99)*x+(42*n+102)*x^2-(12*n^2+12*n+18)*x^3-(40*n^2+96*n+80)*x^4-(44*n^2+76*n+32)*x^5-(16*n^2+16*n)*x^6)*d/dx P(n-2,x) - (18+(12*n+9)*x-24*x^2-(8*n^2+10*n+8)*x^3-(8*n^2-8*n)*x^4)*P(n-1,x) + (9*x+21*x^2-(6*n-12)*x^3-14*n*x^4-8*n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n - 54 + (12*n^2 + 60*n - 144)*x + (12*n - 60)*x^2 - (8*n^3 + 36*n^2 - 32*n - 96)*x^3 - (16*n^3 + 16*n^2 - 64*n - 64)*x^4 - (8*n^3 - 8*n^2 - 16*n)*x^5)*P[n - 2, x] - (27 + (18*n + 99)*x + (42*n + 102)*x^2 - (12*n^2 + 12*n + 18)*x^3 - (40*n^2 + 96*n + 80)*x^4 - (44*n^2 + 76*n + 32)*x^5 - (16*n^2 + 16*n)*x^6)*D[P[n - 2, x], x] - (18 + (12*n + 9)*x - 24*x^2 - (8*n^2 + 10*n + 8)*x^3 - (8*n^2 - 8*n)*x^4)*P[n - 1, x] + (9*x + 21*x^2 - (6*n - 12)*x^3 - 14*n*x^4 - 8*n*x^5)*D[P[n - 1, x], x])/(-3*x + 2*n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((36*n - 54 + (12*n^2 + 60*n - 144)*x + (12*n - 60)*x^2 - (8*n^3 + 36*n^2 - 32*n - 96)*x^3 - (16*n^3 + 16*n^2 - 64*n - 64)*x^4 - (8*n^3 - 8*n^2 - 16*n)*x^5)*v[i - 2] - (27 + (18*n + 99)*x + (42*n + 102)*x^2 - (12*n^2 + 12*n + 18)*x^3 - (40*n^2 + 96*n + 80)*x^4 - (44*n^2 + 76*n + 32)*x^5 - (16*n^2 + 16*n)*x^6)*deriv(v[i - 2], x) - (18 + (12*n + 9)*x - 24*x^2 - (8*n^2 + 10*n + 8)*x^3 - (8*n^2 - 8*n)*x^4)*v[i - 1] + (9*x + 21*x^2 - (6*n - 12)*x^3 - 14*n*x^4 - 8*n*x^5)*deriv(v[i - 1], x))/(-3*x + 2*n*x^3)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.