Real-rooted atlas

A143495 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-2 - n - 4\,n x - 2\,n x^{2} - 2\,n^{2} x - 3\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{2} + 2\,x^{3} + n x^{2} + 3\,n x^{3} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(3 + 2\,x + 2\,n + 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x^{2} - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2167
31142742
4124111108244
51362917004051433
61506152640390514588669
7166114075602040520202510354477
8184193218200771051426329937917496356829

All 15 rows

Technical details
Node id
n-ca1c958ca8cc7d5a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-aaf223662d55759d

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: 1,  14,   27;
   4: 1,  24,  111,   108;
   5: 1,  36,  291,   700,    405;
   6: 1,  50,  615,  2640,   3905,    1458;
   7: 1,  66, 1140,  7560,  20405,   20202,    5103;
   8: 1,  84, 1932, 18200,  77105,  142632,   99379,   17496;
   9: 1, 104, 3066, 38808, 236355,  697788,  931399,  471800,   59049;
  10: 1, 126, 4626, 75600, 624435, 2683170, 5815915, 5794080, 2182149, 196830;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(n+2+(2*n^2+4*n)*x+(n^3+3*n^2+2*n)*x^2)*P(n-2,x) + ((n+2)*x^2+(n^2+3*n+2)*x^3)*d/dx P(n-2,x) + (2*n+3+(2*n^2+4*n+2)*x)*P(n-1,x) - (n+1)*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 2, x] + ((n + 2)*x^2 + (n^2 + 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n + 3 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x] - (n + 1)*x^2*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(n + 2 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*v[i - 2] + ((n + 2)*x^2 + (n^2 + 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n + 3 + (2*n^2 + 4*n + 2)*x)*v[i - 1] - (n + 1)*x^2*deriv(v[i - 1], x))/(n + 1)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.