Real-rooted atlas

A134991 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x + n\right)\,P_{n}(x) = \left(4\,x + 2\,x^{2} - 2\,x^{3} - 2\,n - n x + 5\,n x^{2} - 2\,n x^{4} - 2\,n^{2} x + 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{2} - 4\,x^{4} - 2\,x^{5} - n x + 4\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 - 9\,x - 4\,x^{2} + 2\,x^{3} + 3\,n + n x - 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x - 3\,x^{2} - 2\,x^{3} + n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
311401566
426235420105786
55710364270504094511348
6120395530268787506930010395192788
724713896177765819280154192510810801351353769328
8502463719319206849535221321103247744518918900202702583383808

All 15 rows

Technical details
Node id
n-ceb547fe03cf0e38 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d611ddd898cc65fb

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:    4,      3;
   3:   11,     40,       15;
   4:   26,    235,      420,       105;
   5:   57,   1036,     4270,      5040,        945;
   6:  120,   3955,    30268,     78750,      69300,      10395;
   7:  247,  13896,   177765,    819280,    1541925,    1081080,      135135;
   8:  502,  46371,   931920,   6849535,   22132110,   32477445,    18918900,     2027025;
   9: 1013, 149644,  4535916,  49975640,  249659410,  615074460,   737837100,   367567200,   34459425;
  10: 2036, 472219, 20979816, 332640308, 2415813400, 9000561570, 17824967160, 18056738700, 7856748900, 654729075;
%F Formula
The row polynomials satisfy (n-1-2*x)*P(n,x) = -(2*n+(2*n^2+n-4)*x+(n^3-5*n-2)*x^2-(3*n^2-2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2)*x^2+(n^2-4*n)*x^3-(3*n-4)*x^4+2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2+n-9)*x-(5*n+4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x+(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 + n - 4)*x + (n^3 - 5*n - 2)*x^2 - (3*n^2 - 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2)*x^2 + (n^2 - 4*n)*x^3 - (3*n - 4)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 + n - 9)*x - (5*n + 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x + (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 - 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 + n - 4)*x + (n^3 - 5*n - 2)*x^2 - (3*n^2 - 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2)*x^2 + (n^2 - 4*n)*x^3 - (3*n - 4)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 + n - 9)*x - (5*n + 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x + (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 - 2*x)); 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. [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.