Real-rooted atlas

A290598 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(28 + 147\,x + 263\,x^{2} + 198\,x^{3} + 54\,x^{4} - 12\,n - 87\,n x - 183\,n x^{2} - 153\,n x^{3} - 45\,n x^{4} + 9\,n^{2} x + 27\,n^{2} x^{2} + 27\,n^{2} x^{3} + 9\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(12 + 108\,x + 306\,x^{2} + 390\,x^{3} + 234\,x^{4} + 54\,x^{5} - 18\,n x - 72\,n x^{2} - 108\,n x^{3} - 72\,n x^{4} - 18\,n x^{5}\right)\,P_{n-1}'(x) + \left(9\,x + 45\,x^{2} + 90\,x^{3} + 90\,x^{4} + 45\,x^{5} + 9\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2459
3287043141
4280105012905213141
53640182003354027092811390585
65824036400089440010836806490401540013203361
711065608299200254904004117984036995280175561143442915134070309
824344320213012800785104320158542384018990910401351820778530208910884911736477497181

All 15 rows

Technical details
Node id
n-0f37266207fa0305 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-64e5a7bd48c32af1

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:           4,            5;
   3:          28,           70,           43;
   4:         280,         1050,         1290,           521;
   5:        3640,        18200,        33540,         27092,          8113;
   6:       58240,       364000,       894400,       1083680,        649040,        154001;
   7:     1106560,      8299200,     25490400,      41179840,      36995280,      17556114,       3442915;
   8:    24344320,    213012800,    785104320,    1585423840,    1899091040,    1351820778,     530208910,      88491173;
   9:   608608000,   6086080000,  26170144000,   63416953600,   94954552000,   90121385200,   53020891000,   17698234600,   2568458401;
  10: 17041024000, 191711520000, 942125184000, 2663512051200, 4785709420800, 5677647267600, 4453754844000, 2229977559600, 647251517052, 83033591453;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(12*n-28-(9*n^2-87*n+147)*x-(27*n^2-183*n+263)*x^2-(27*n^2-153*n+198)*x^3-(9*n^2-45*n+54)*x^4)*P(n-1,x) + (12-(18*n-108)*x-(72*n-306)*x^2-(108*n-390)*x^3-(72*n-234)*x^4-(18*n-54)*x^5)*d/dx P(n-1,x) + (9*x+45*x^2+90*x^3+90*x^4+45*x^5+9*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[(-(12*n - 28 - (9*n^2 - 87*n + 147)*x - (27*n^2 - 183*n + 263)*x^2 - (27*n^2 - 153*n + 198)*x^3 - (9*n^2 - 45*n + 54)*x^4)*P[n - 1, x] + (12 - (18*n - 108)*x - (72*n - 306)*x^2 - (108*n - 390)*x^3 - (72*n - 234)*x^4 - (18*n - 54)*x^5)*D[P[n - 1, x], x] + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*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] = (-(12*n - 28 - (9*n^2 - 87*n + 147)*x - (27*n^2 - 183*n + 263)*x^2 - (27*n^2 - 153*n + 198)*x^3 - (9*n^2 - 45*n + 54)*x^4)*v[i - 1] + (12 - (18*n - 108)*x - (72*n - 306)*x^2 - (108*n - 390)*x^3 - (72*n - 234)*x^4 - (18*n - 54)*x^5)*deriv(v[i - 1], x) + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*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.