Real-rooted atlas

A391533 → Pₙ + Pₙ₊₁ → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8 + 36\,x + 54\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
08365427125
185212613554375
28682303873241081125
388436684710987562163375
48100534157927922952172843210125
581167342647595085367632388886430375
681329664115112442043624704191528640172891125
78148123060471947442924655766856046944190083456273375
88164152685073156881872151424199712184064112896414726912820125
981801854115594858214500831516850256058348848102426726489856138242460375
108196221415267717002421726051841132896158860816480001229312624384193536276487381125
11821226061969510223438557210895282343264385440048252164525312308300814423044147205529622143375
1282283030249071416245900401860672452232085409281253401614175744121336327608320329932888473611059266430125
1382443486309671914388732883040752824366417585568296158723924377640485120318755841851596874833921880064221184199290375
14826039743793925337212561644787328143251683407289664787008984755201189726721128458248226713644515328168468483981312442368597871125
Technical details
Node id
n-ad0bc0d16494c943 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-618c2d9a062c7264

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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: 8,  36,   54,    27;
  1: 8,  52,  126,   135,    54;
  2: 8,  68,  230,   387,   324,    108;
  3: 8,  84,  366,   847,  1098,    756,    216;
  4: 8, 100,  534,  1579,  2792,   2952,   1728,    432;
  5: 8, 116,  734,  2647,  5950,   8536,   7632,   3888,    864;
  6: 8, 132,  966,  4115, 11244,  20436,  24704,  19152,   8640,   1728;
  7: 8, 148, 1230,  6047, 19474,  42924,  65576,  68560,  46944,  19008,   3456;
  8: 8, 164, 1526,  8507, 31568,  81872, 151424, 199712, 184064, 112896,  41472,  6912;
  9: 8, 180, 1854, 11559, 48582, 145008, 315168, 502560, 583488, 481024, 267264, 89856, 13824;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1+3*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 36*x + 54*x^2 + 27*x^3.
%t Mathematica
P[0, x_] := 8 + 36*x + 54*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 3*x + 2*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] = 8 + 36*x + 54*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 3*x + 2*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

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.