Real-rooted atlas

A049029 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-3 - 3\,x + 2\,x^{2} + 2\,x^{3} + 4\,n + 7\,n x + 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
25611
34510561211
4585201022958715761
5994544955760205700515996203921
620884511680202607780290569516159803588918865211
752211253475426596230295141881880117493845518187609509641456909811
815141262511679349503855423285706089709577488321055095995240185968758529052831627230711201

All 15 rows

Technical details
Node id
n-9e325ca0f01cd501 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9fa03cb4ba12e174

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:            5,             6;
   3:           45,           105,            61;
   4:          585,          2010,          2295,            871;
   5:         9945,         44955,         76020,          57005,          15996;
   6:       208845,       1168020,       2607780,        2905695,        1615980,         358891;
   7:      5221125,      34754265,      96230295,      141881880,      117493845,       51818760,        9509641;
   8:    151412625,    1167934950,    3855423285,     7060897095,     7748832105,     5095995240,     1859687585,      290528316;
   9:   4996616625,   43797929175,  167750040150,   366702807555,   500439275595,   436613354415,   237833143380,    73956957975,   10051973371;
  10: 184874815125, 1814219807400, 7903812366450, 20065028163075, 32712811130475, 35520939320265, 25689602239560, 11933258109615, 3230778665835, 388433817091;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n-3+(7*n-3)*x+(2*n+2)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx 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 - 3 + (7*n - 3)*x + (2*n + 2)*x^2 - (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(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 - 3 + (7*n - 3)*x + (2*n + 2)*x^2 - (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.