Real-rooted atlas

A120910 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 10\,x + 7\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
192130
227126147300
381567132310293000
42432268793812348720330000
57298505396909261010804550421300000
62187306181786055556609724059075783529473000000
765611071637501412917215680683595295697411887247062930000000

All 15 rows

Technical details
Node id
n-7c877ed9ba2e6414 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5184fe74b2189306

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     3;
  1:     9,      21;
  2:    27,     126,      147;
  3:    81,     567,     1323,     1029;
  4:   243,    2268,     7938,    12348,      7203;
  5:   729,    8505,    39690,    92610,    108045,     50421;
  6:  2187,   30618,   178605,   555660,    972405,    907578,    352947;
  7:  6561,  107163,   750141,  2917215,   6806835,   9529569,   7411887,   2470629;
  8: 19683,  367416,  3000564, 14002632,  40841010,  76236552,  88942644,  59295096,  17294403;
  9: 59049, 1240029, 11573604, 63011844, 220541454, 514596726, 800483796, 800483796, 466948881, 121060821;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3+10*x+7*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((3 + 10*x + 7*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((3 + 10*x + 7*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.