Real-rooted atlas

A113214 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 8\,n x + n x^{3} - 3\,n^{2} + 2\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(16 - 36\,x - 12\,x^{2} + 6\,x^{3} + 2\,x^{4} - 28\,n + 12\,n x + 6\,n x^{2} + 14\,n^{2} - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x - 22\,x^{2} + 2\,x^{3} + 6\,x^{4} + x^{5} - 12\,n x + 5\,n x^{2} + 6\,n x^{3} + n x^{4} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
2612321
342824460
403075405150
5012114156606348
60084308280847763
7002432868045611281608

All 15 rows

Technical details
Node id
n-ccb1508b738b30b0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0d29bcfb6cb34365

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: 4,  2;
  2: 6, 12,   3;
  3: 4, 28,  24,   4;
  4: 0, 30,  75,  40,   5;
  5: 0, 12, 114, 156,  60,    6;
  6: 0,  0,  84, 308, 280,   84,    7;
  7: 0,  0,  24, 328, 680,  456,  112,    8;
  8: 0,  0,   0, 180, 945, 1314,  693,  144,   9;
  9: 0,  0,   0,  40, 760, 2250, 2310, 1000, 180, 10;
%F Formula
The row polynomials satisfy (n^3-3*n^2+2*n+(2*n^2-8*n)*x+n^2*x^2+n*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(12*n-36)*x+(2*n^2-6*n+12)*x^2-6*x^3-2*x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2+5*n-22)*x^2+(6*n+2)*x^3+(n+6)*x^4+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[(-(2*n^3 - 14*n^2 + 28*n - 16 - (12*n - 36)*x + (2*n^2 - 6*n + 12)*x^2 - 6*x^3 - 2*x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 + 5*n - 22)*x^2 + (6*n + 2)*x^3 + (n + 6)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n + (2*n^2 - 8*n)*x + n^2*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; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (12*n - 36)*x + (2*n^2 - 6*n + 12)*x^2 - 6*x^3 - 2*x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 + 5*n - 22)*x^2 + (6*n + 2)*x^3 + (n + 6)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n + (2*n^2 - 8*n)*x + n^2*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

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.