Real-rooted atlas

A094638 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2181322
31156070146
41241714744461116
51353851855406032769612
6148750548020525379602725292016
716313231354576020237888387128253296969264
8180217029540230125106946029190004291560260222411144160
9199336658674603141386255115545684380625965149108829288160138915360
1011204995108360141739511905992656454052356227605277170046656885283584577601866568320
11114371501887603056625325074752339377041144682110373982866477714928729233617536474057792026899896960
12116899333133906148065805995967312984914650991422206284187346226802489612135244413613686690124867375532160413960682240
13119513455499785116723751847038052058367025164459702559446775967238570997036010879108583202005073843760215999674675210242087206406776069126400
14122417836770224211070863963479525314737428519934249123740257170651971588482864749798543093619935414180864349782765864483617218290816016583626886400117570826598400
Technical details
Node id
n-96631c9ffb8654db · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c5be635e87f1100d

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: 1,  3;
  2: 1,  8,   13;
  3: 1, 15,   60,    70;
  4: 1, 24,  171,   474,    446;
  5: 1, 35,  385,  1855,   4060,    3276;
  6: 1, 48,  750,  5480,  20525,   37960,    27252;
  7: 1, 63, 1323, 13545,  76020,  237888,   387128,   253296;
  8: 1, 80, 2170, 29540, 230125, 1069460,  2919000,  4291560,  2602224;
  9: 1, 99, 3366, 58674, 603141, 3862551, 15545684, 38062596, 51491088, 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) = 1 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 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] = 1 + 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.