Real-rooted atlas

A286724 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 61\,x + 115\,x^{2} + 88\,x^{3} + 24\,x^{4} - 4\,n - 36\,n x - 80\,n x^{2} - 68\,n x^{3} - 20\,n x^{4} + 4\,n^{2} x + 12\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4 + 44\,x + 132\,x^{2} + 172\,x^{3} + 104\,x^{4} + 24\,x^{5} - 8\,n x - 32\,n x^{2} - 48\,n x^{3} - 32\,n x^{4} - 8\,n x^{5}\right)\,P_{n-1}'(x) + \left(4\,x + 20\,x^{2} + 40\,x^{3} + 40\,x^{4} + 20\,x^{5} + 4\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
38241749
448216306139709
5384230448964448147313505
6384028800816001112007365019091318181
74608041472014688002668800265140013745522917938916145
8645120677376028788480653856008661240067353048285957145129307289283429

All 15 rows

Technical details
Node id
n-cadc1db6e74f2494 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1f17426991eee5e3

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 -1)
  • 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:         1;
   2:         2,          3;
   3:         8,         24,          17;
   4:        48,        216,         306,         139;
   5:       384,       2304,        4896,        4448,        1473;
   6:      3840,      28800,       81600,      111200,       73650,        19091;
   7:     46080,     414720,     1468800,     2668800,     2651400,      1374552,      291793;
   8:    645120,    6773760,    28788480,    65385600,    86612400,     67353048,    28595714,     5129307;
   9:  10321920,  123863040,   614154240,  1673871360,  2771596800,   2873730048,  1830125696,   656551296,   101817089;
  10: 185794560, 2508226560, 14213283840, 45194526720, 89799736320, 116386066944, 98826787584, 53180654976, 16494368418, 2250495523;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(4*n-10-(4*n^2-36*n+61)*x-(12*n^2-80*n+115)*x^2-(12*n^2-68*n+88)*x^3-(4*n^2-20*n+24)*x^4)*P(n-1,x) + (4-(8*n-44)*x-(32*n-132)*x^2-(48*n-172)*x^3-(32*n-104)*x^4-(8*n-24)*x^5)*d/dx P(n-1,x) + (4*x+20*x^2+40*x^3+40*x^4+20*x^5+4*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 10 - (4*n^2 - 36*n + 61)*x - (12*n^2 - 80*n + 115)*x^2 - (12*n^2 - 68*n + 88)*x^3 - (4*n^2 - 20*n + 24)*x^4)*P[n - 1, x] + (4 - (8*n - 44)*x - (32*n - 132)*x^2 - (48*n - 172)*x^3 - (32*n - 104)*x^4 - (8*n - 24)*x^5)*D[P[n - 1, x], x] + (4*x + 20*x^2 + 40*x^3 + 40*x^4 + 20*x^5 + 4*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n - 10 - (4*n^2 - 36*n + 61)*x - (12*n^2 - 80*n + 115)*x^2 - (12*n^2 - 68*n + 88)*x^3 - (4*n^2 - 20*n + 24)*x^4)*v[i - 1] + (4 - (8*n - 44)*x - (32*n - 132)*x^2 - (48*n - 172)*x^3 - (32*n - 104)*x^4 - (8*n - 24)*x^5)*deriv(v[i - 1], x) + (4*x + 20*x^2 + 40*x^3 + 40*x^4 + 20*x^5 + 4*x^6)*deriv(deriv(v[i - 1], x), x))/(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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.