Real-rooted atlas

A161198 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 + 4\,x + 5\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
23912
3157899192
4105741170712873840
5945824426478371881930592160
6103951067494314068591948445513281852580480
713513515852427636509193662362730384920312874623551582575360
82027025266164651478377174508602298162486918780739395201030791309458152972712960

All 15 rows

Technical details
Node id
n-6069d893b32d8be9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b7e8d702e00ffa0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         1;
   2:         3,           9;
   3:        15,          78,          99;
   4:       105,         741,        1707,         1287;
   5:       945,        8244,       26478,        37188,        19305;
   6:     10395,      106749,      431406,       859194,       844551,        328185;
   7:    135135,     1585242,     7636509,     19366236,     27303849,      20312874,       6235515;
   8:   2027025,    26616465,   147837717,    450860229,    816248691,     878073939,     520103079,    130945815;
   9:  34459425,   499101480,  3125419884,  11064891384,  24246013014,   33700976856,   29037452940,  14188449672,   3011753745;
  10: 654729075, 10344413745, 71860514796, 288368433396, 737296531866, 1246468885614, 1394236027260, 995516867268, 411934562955, 75293843625;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(4*n+4)*x+(2*n+5)*x^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[((2*n - 1 + (4*n + 4)*x + (2*n + 5)*x^2)*P[n - 1, 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] = ((2*n - 1 + (4*n + 4)*x + (2*n + 5)*x^2)*v[i - 1])/(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.