Real-rooted atlas

A161198 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2358
315463596
41054576593151536
594552681095810084346530720
6103957023318902225337816920745045737280
71351351068954351078161292446000361312369067567520643840
8202702518331605708339331516219371942025631488614876323785511486475660602880

All 15 rows

Technical details
Node id
n-f6ed131bd69c89a9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cd9385ed58c16152

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,          5;
   3:        15,         46,          35;
   4:       105,        457,         659,          315;
   5:       945,       5268,       10958,        10084,         3465;
   6:     10395,      70233,      189022,       253378,       169207,        45045;
   7:    135135,    1068954,     3510781,      6129244,      6000361,      3123690,       675675;
   8:   2027025,   18331605,    70833933,    151621937,    194202563,    148861487,     63237855,     11486475;
   9:  34459425,  350150760,  1552477356,   3923417656,   6182260374,   6220493976,   3903411788,   1396789320,   218243025;
  10: 654729075, 7376512365, 36850235724, 107146959940, 199854717882, 248016853398, 204795197468, 108510644628, 33479193195, 4583103525;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+4*n*x+(2*n+1)*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*x + (2*n + 1)*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*x + (2*n + 1)*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.