Real-rooted atlas

A152198 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 - 6\,x + 4\,x^{2} - n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(4 + 8\,x^{2} - 16\,x^{3} - 8\,n - 2\,n x - 20\,n x^{2} + 16\,n x^{3} + n^{2} - 4\,n^{2} x + 16\,n^{2} x^{2} + n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,x - 24\,x^{2} - 40\,x^{3} - 32\,x^{4} - 3\,n - 14\,n x - 32\,n x^{2} - 32\,n x^{3} - n^{2} - 6\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
26410
36410
420481684
520481684
65624028864648
75624028864648
8144896192015362564752
9144896192015362564752
1035228808960128007680102433696
1135228808960128007680102433696
128328448345607168076800368644096233280
138328448345607168076800368644096233280
14192023296118272322560501760430080172032163841586304
Technical details
Node id
n-3e0cfe5cd186ce33 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4f4ca2dec4972ed6

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   1;
  2:   6,   4;
  3:   6,   4;
  4:  20,  48,   16;
  5:  20,  48,   16;
  6:  56, 240,  288,   64;
  7:  56, 240,  288,   64;
  8: 144, 896, 1920, 1536, 256;
  9: 144, 896, 1920, 1536, 256;
%F Formula
The row polynomials satisfy (n^2-n-2+(4*n-6)*x+4*x^2)*P(n,x) = (n^3+n^2-8*n+4+(4*n^3-4*n^2-2*n)*x+(16*n^2-20*n+8)*x^2+(16*n-16)*x^3)*P(n-1,x) - (n^2+3*n+(6*n^2+14*n+6)*x+(8*n^2+32*n+24)*x^2+(32*n+40)*x^3+32*x^4)*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[((n^3 + n^2 - 8*n + 4 + (4*n^3 - 4*n^2 - 2*n)*x + (16*n^2 - 20*n + 8)*x^2 + (16*n - 16)*x^3)*P[n - 1, x] - (n^2 + 3*n + (6*n^2 + 14*n + 6)*x + (8*n^2 + 32*n + 24)*x^2 + (32*n + 40)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (4*n - 6)*x + 4*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] = ((n^3 + n^2 - 8*n + 4 + (4*n^3 - 4*n^2 - 2*n)*x + (16*n^2 - 20*n + 8)*x^2 + (16*n - 16)*x^3)*v[i - 1] - (n^2 + 3*n + (6*n^2 + 14*n + 6)*x + (8*n^2 + 32*n + 24)*x^2 + (32*n + 40)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(n^2 - n - 2 + (4*n - 6)*x + 4*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

Same fingerprint ?

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.