Real-rooted atlas

A133081 → reverse (shift +1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x + x^{2} - 10\,n + 4\,n x + 4\,n^{2}\right)\,P_{n}(x) = \left(12\,x - 9\,x^{2} - 17\,x^{3} + 5\,x^{4} - 38\,n x - 24\,n x^{2} + 34\,n x^{3} - 4\,n x^{4} + 32\,n^{2} x + 44\,n^{2} x^{2} - 16\,n^{2} x^{3} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(3\,x^{2} - 2\,x^{3} - 5\,x^{4} + 2\,x^{5} - 8\,n x^{2} - 8\,n x^{3} + 8\,n x^{4} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11315
2000415
3000057113
40000000819
500000000911121
60000000000012113
70000000000001315129
800000000000000016117
900000000000000001719137
10000000000000000000020121
11000000000000000000002123145
120000000000000000000000024125
130000000000000000000000002527153
1400000000000000000000000000028129
Technical details
Node id
n-e31c5b031b3081bb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2f306733f1e6c36c

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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, 3, 1;
  2: 0, 0, 0, 4, 1;
  3: 0, 0, 0, 0, 5, 7, 1;
  4: 0, 0, 0, 0, 0, 0, 0, 8, 1;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 9, 11, 1;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0, 12,  1;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0, 13, 15, 1;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 0, 16,  1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 0,  0, 17, 19, 1;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6+(4*n-6)*x+x^2)*P(n,x) = -((8*n^3-32*n^2+38*n-12)*x+(16*n^3-44*n^2+24*n+9)*x^2+(16*n^2-34*n+17)*x^3+(4*n-5)*x^4)*P(n-1,x) + ((4*n^2-8*n+3)*x^2+(8*n^2-8*n-2)*x^3+(8*n-5)*x^4+2*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[(-((8*n^3 - 32*n^2 + 38*n - 12)*x + (16*n^3 - 44*n^2 + 24*n + 9)*x^2 + (16*n^2 - 34*n + 17)*x^3 + (4*n - 5)*x^4)*P[n - 1, x] + ((4*n^2 - 8*n + 3)*x^2 + (8*n^2 - 8*n - 2)*x^3 + (8*n - 5)*x^4 + 2*x^5)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (4*n - 6)*x + x^2)]];
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] = (-((8*n^3 - 32*n^2 + 38*n - 12)*x + (16*n^3 - 44*n^2 + 24*n + 9)*x^2 + (16*n^2 - 34*n + 17)*x^3 + (4*n - 5)*x^4)*v[i - 1] + ((4*n^2 - 8*n + 3)*x^2 + (8*n^2 - 8*n - 2)*x^3 + (8*n - 5)*x^4 + 2*x^5)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (4*n - 6)*x + x^2)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.