Real-rooted atlas

A274498 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 2\,x - 4\,x^{2} + 2\,n - 5\,n x + 8\,n x^{2} + 2\,n^{2} x - 5\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-48 + 96\,x - 48\,x^{2} + 30\,n - 153\,n x + 114\,n x^{2} - 4\,n^{2} + 60\,n^{2} x - 101\,n^{2} x^{2} + 6\,n^{2} x^{3} - 6\,n^{3} x + 29\,n^{3} x^{2} - 8\,n^{3} x^{3} - 2\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-6 + 6\,x + 24\,x^{2} - 24\,x^{3} + 2\,n - 18\,n x - 18\,n x^{2} + 52\,n x^{3} + 4\,n^{2} x - 10\,n^{2} x^{2} - 34\,n^{2} x^{3} + 4\,n^{2} x^{4} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
266
31818
436144180
5108432540
6162129619443402
74863888583210206
86487776233282073652488
91944233286998462208157464
10243038880174960311040194400721710
1172901166405248809331205832002165130
12874817496010497602799360349920016796169211644
13262445248803149280839808010497600503884827634932
1430618734832551124019595520367416003527193613716864111602610
Technical details
Node id
n-1971eb333384ff5d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4a9146a6908256f6

Observed on the stored rows ?

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    0;
  1:    0;
  2:    6;
  3:   18;
  4:   36,   144;
  5:  108,   432;
  6:  162,  1296,  1944;
  7:  486,  3888,  5832;
  8:  648,  7776, 23328, 20736;
  9: 1944, 23328, 69984, 62208;
%F Formula
The row polynomials satisfy (2*n-4+(2*n^2-5*n+2)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n,x) = -(4*n^2-30*n+48+(6*n^3-60*n^2+153*n-96)*x+(2*n^4-29*n^3+101*n^2-114*n+48)*x^2-(2*n^4-8*n^3+6*n^2)*x^3)*P(n-1,x) + (2*n-6+(4*n^2-18*n+6)*x+(2*n^3-10*n^2-18*n+24)*x^2+(2*n^3-34*n^2+52*n-24)*x^3-(4*n^3-4*n^2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 30*n + 48 + (6*n^3 - 60*n^2 + 153*n - 96)*x + (2*n^4 - 29*n^3 + 101*n^2 - 114*n + 48)*x^2 - (2*n^4 - 8*n^3 + 6*n^2)*x^3)*P[n - 1, x] + (2*n - 6 + (4*n^2 - 18*n + 6)*x + (2*n^3 - 10*n^2 - 18*n + 24)*x^2 + (2*n^3 - 34*n^2 + 52*n - 24)*x^3 - (4*n^3 - 4*n^2)*x^4)*D[P[n - 1, x], x])/(2*n - 4 + (2*n^2 - 5*n + 2)*x + (n^3 - 5*n^2 + 8*n - 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] = 0; v[2] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 30*n + 48 + (6*n^3 - 60*n^2 + 153*n - 96)*x + (2*n^4 - 29*n^3 + 101*n^2 - 114*n + 48)*x^2 - (2*n^4 - 8*n^3 + 6*n^2)*x^3)*v[i - 1] + (2*n - 6 + (4*n^2 - 18*n + 6)*x + (2*n^3 - 10*n^2 - 18*n + 24)*x^2 + (2*n^3 - 34*n^2 + 52*n - 24)*x^3 - (4*n^3 - 4*n^2)*x^4)*deriv(v[i - 1], x))/(2*n - 4 + (2*n^2 - 5*n + 2)*x + (n^3 - 5*n^2 + 8*n - 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.