Real-rooted atlas

A008277 → θ+1 → reverse → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 12\,x - 30\,x^{2} - 2\,n x + 11\,n x^{2} - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(x^{2} - 5\,x^{3} + n x^{3}\right)\,P_{n-4}'(x) + \left(4 - 28\,x + 25\,x^{2} + 6\,n x - 15\,n x^{2} + 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-3\,x^{2} + 6\,x^{3} - 2\,n x^{3}\right)\,P_{n-3}'(x) + \left(-6 + 20\,x - 3\,x^{2} - 6\,n x + 4\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3\,x^{2} - x^{3} + n x^{3}\right)\,P_{n-2}'(x) + \left(4 - 4\,x + 2\,n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 6 + 2\,x\)
\(P_{4}(x) = 18 + 28\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3628
41828349
540150904284
67552081024851658
712614004200361263069974
819631921575027216144901524762375

All 15 rows

Technical details
Node id
n-548558540166a2f0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b5d5cce567e2d05a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:   0;
   2:   1;
   3:   6,     2;
   4:  18,    28,      3;
   5:  40,   150,     90,      4;
   6:  75,   520,    810,    248,       5;
   7: 126,  1400,   4200,   3612,     630,      6;
   8: 196,  3192,  15750,  27216,   14490,   1524,      7;
   9: 288,  6468,  47628, 139020,  155400,  54450,   3570,    8;
  10: 405, 12000, 123480, 547848, 1063125, 818520, 195930, 8176, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-12)*x+(n^2-11*n+30)*x^2)*P(n-4,x) + (x^2+(n-5)*x^3)*d/dx P(n-4,x) + (4+(6*n-28)*x+(2*n^2-15*n+25)*x^2)*P(n-3,x) - (3*x^2+(2*n-6)*x^3)*d/dx P(n-3,x) - (6+(6*n-20)*x+(n^2-4*n+3)*x^2)*P(n-2,x) + (3*x^2+(n-1)*x^3)*d/dx P(n-2,x) + (4+(2*n-4)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 6 + 2*x, P(4,x) = 18 + 28*x + 3*x^2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 6 + 2*x;
P[4, x_] := 18 + 28*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 12)*x + (n^2 - 11*n + 30)*x^2)*P[n - 4, x] + (x^2 + (n - 5)*x^3)*D[P[n - 4, x], x] + (4 + (6*n - 28)*x + (2*n^2 - 15*n + 25)*x^2)*P[n - 3, x] - (3*x^2 + (2*n - 6)*x^3)*D[P[n - 3, x], x] - (6 + (6*n - 20)*x + (n^2 - 4*n + 3)*x^2)*P[n - 2, x] + (3*x^2 + (n - 1)*x^3)*D[P[n - 2, x], x] + (4 + (2*n - 4)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 6 + 2*x; v[4] = 18 + 28*x + 3*x^2; for(n = 5, N, my(i = n); v[i] = -(1 + (2*n - 12)*x + (n^2 - 11*n + 30)*x^2)*v[i - 4] + (x^2 + (n - 5)*x^3)*deriv(v[i - 4], x) + (4 + (6*n - 28)*x + (2*n^2 - 15*n + 25)*x^2)*v[i - 3] - (3*x^2 + (2*n - 6)*x^3)*deriv(v[i - 3], x) - (6 + (6*n - 20)*x + (n^2 - 4*n + 3)*x^2)*v[i - 2] + (3*x^2 + (n - 1)*x^3)*deriv(v[i - 2], x) + (4 + (2*n - 4)*x)*v[i - 1] - x^2*deriv(v[i - 1], 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. [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.