Real-rooted atlas

A094638 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-n - 2\,n x - n x^{2} - 2\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{2} x^{3} - n^{3} x^{2} - 2\,n^{3} x^{3} - n^{3} x^{4}\right)\,P_{n-2}(x) + \left(-1 - 3\,x - 3\,x^{2} - x^{3} + 2\,n + 4\,n x + 2\,n x^{2} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
24151332
374910670232
4111244918004461872
516265166048716616327616704
622503456520901504576017227252163872
72987510843711412650485557526001042532961757088
8371424230862050161084909347945665468646530904260222420473920
94621994515652092637045261675247147854544825538447715044829288160257872320
10563255825451199142110124426652280726630102569304882811127784929842294883584577603493635840
1167465314278025490522933379922768524912113095944394614158105872032641599766912813495193856474057792050686283520
127964602358735074432714650316921442444711100603226144806807560366935017059949021624464360714419801541260867375532160784332218880
139287493748169560837161675228190936016716156594168988521501114355759430881358501009944289711526041639687832343523097027176192102420872064012898301068800
141061159957612117189627343584241485686043749954759283378741415481212008871574586854272909042551073288909651779361786192681189708610085144813542656016583626886400224680258252800
Technical details
Node id
n-d79ce3e611e817f6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cb23b54495d78fc3

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2,    3;
  2:  4,   15,    13;
  3:  7,   49,   106,     70;
  4: 11,  124,   491,    800,     446;
  5: 16,  265,  1660,   4871,    6616,     3276;
  6: 22,  503,  4565,  20901,   50457,    60172,    27252;
  7: 29,  875, 10843,  71141,  265048,   555752,   600104,   253296;
  8: 37, 1424, 23086, 205016, 1084909,  3479456,  6546864,  6530904,  2602224;
  9: 46, 2199, 45156, 520926, 3704526, 16752471, 47854544, 82553844, 77150448, 29288160;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(n+(2*n^2+2*n)*x+(n^3+4*n^2+n)*x^2+(2*n^3+2*n^2)*x^3+n^3*x^4)*P(n-2,x) + (2*n-1+(2*n^2+4*n-3)*x+(4*n^2+2*n-3)*x^2+(2*n^2-1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 + 2*n)*x + (n^3 + 4*n^2 + n)*x^2 + (2*n^3 + 2*n^2)*x^3 + n^3*x^4)*P[n - 2, x] + (2*n - 1 + (2*n^2 + 4*n - 3)*x + (4*n^2 + 2*n - 3)*x^2 + (2*n^2 - 1)*x^3)*P[n - 1, x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (2*n^2 + 2*n)*x + (n^3 + 4*n^2 + n)*x^2 + (2*n^3 + 2*n^2)*x^3 + n^3*x^4)*v[i - 2] + (2*n - 1 + (2*n^2 + 4*n - 3)*x + (4*n^2 + 2*n - 3)*x^2 + (2*n^2 - 1)*x^3)*v[i - 1])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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. [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.