Real-rooted atlas

A086915 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2 + 2\,x\right)\,P_{n}(x) = \left(6 + 24\,x + 28\,x^{2} + 12\,x^{3} + 2\,x^{4} - 2\,n - 13\,n x - 23\,n x^{2} - 15\,n x^{3} - 3\,n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(2 + 16\,x + 40\,x^{2} + 44\,x^{3} + 22\,x^{4} + 4\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
14812
2124844104
3482885283041168
4240192052806080251216032
514401440052800912007536024064259264
6100801209605544001276800158256010106882615364817024
78064011289606209280178752002954112028299264146460163173888100954368

All 15 rows

Technical details
Node id
n-9380a7775ada7f60 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b274021ddeb555d2

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 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:       2;
  1:       4,         8;
  2:      12,        48,        44;
  3:      48,       288,       528,        304;
  4:     240,      1920,      5280,       6080,       2512;
  5:    1440,     14400,     52800,      91200,      75360,       24064;
  6:   10080,    120960,    554400,    1276800,    1582560,     1010688,      261536;
  7:   80640,   1128960,   6209280,   17875200,   29541120,    28299264,    14646016,     3173888;
  8:  725760,  11612160,  74511360,  257402880,  531740160,   679182336,   527256576,   228519936,   42483968;
  9: 7257600, 130636800, 958003200, 3861043200, 9571322880, 15281602560, 15817697280, 10283397120, 3823557120, 621159424;
%F Formula
The row polynomials satisfy (2+2*x)*P(n,x) = -(2*n-6-(n^2-13*n+24)*x-(3*n^2-23*n+28)*x^2-(3*n^2-15*n+12)*x^3-(n^2-3*n+2)*x^4)*P(n-1,x) + (2-(2*n-16)*x-(8*n-40)*x^2-(12*n-44)*x^3-(8*n-22)*x^4-(2*n-4)*x^5)*d/dx P(n-1,x) + (x+5*x^2+10*x^3+10*x^4+5*x^5+x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 6 - (n^2 - 13*n + 24)*x - (3*n^2 - 23*n + 28)*x^2 - (3*n^2 - 15*n + 12)*x^3 - (n^2 - 3*n + 2)*x^4)*P[n - 1, x] + (2 - (2*n - 16)*x - (8*n - 40)*x^2 - (12*n - 44)*x^3 - (8*n - 22)*x^4 - (2*n - 4)*x^5)*D[P[n - 1, x], x] + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*D[P[n - 1, x], {x, 2}])/(2 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 6 - (n^2 - 13*n + 24)*x - (3*n^2 - 23*n + 28)*x^2 - (3*n^2 - 15*n + 12)*x^3 - (n^2 - 3*n + 2)*x^4)*v[i - 1] + (2 - (2*n - 16)*x - (8*n - 40)*x^2 - (12*n - 44)*x^3 - (8*n - 22)*x^4 - (2*n - 4)*x^5)*deriv(v[i - 1], x) + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*deriv(deriv(v[i - 1], x), x))/(2 + 2*x)); 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.