Real-rooted atlas

A090582 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 2\,x - x^{2} - 4\,x^{3} - 2\,x^{4} + n + 5\,n x + 9\,n x^{2} + 7\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(-x - 5\,x^{2} - 9\,x^{3} - 7\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
26181337
32410815875365
4120720159015305414501
57205400159602322016622468366605
6504045360168000327600354606201978472931149877
7403204233601884960461160066936245762988272487854583522687565
836288043545602264976066679200121491720140268240100203270404928907087261503589781
93628800489888002915136001003363200220097304031905090003055965720186491754065794494210224756312420052205
10399168005987520004011638400158070528004055862888070800927120851471940006965533080037093887006116118349381622632573336947795317
114790016007903526400588772800002613552480007680245126401568636435520227200339872023334938352001665417069624786612472668221291822798280915675959972186170765
1262270208001120863744009190044864004537941408000150286033497603516336375552059597732113920737206614144006604952686332041799405407760177358901401504530383962650526858348381319727684645461
13871782912001699976678400152126118144008271040377600030480436546560080392550958720015611602406400002259884096870400243844185200616019369465441326001100932640360280424010595595860991795810332621064134297044311039636939221805
141307674368000274611617280002663296796160001580978310912000641696880464640018837594135360000412429095029952006840972670371840086382902367281520826322417633512805894327988385200030402806419767600107192299456242062312402554523898230283190977853408406422098722357
Technical details
Node id
n-a53c7a97bbddcdeb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d4e4e266d9f48188

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:       6,       18,        13;
  3:      24,      108,       158,         75;
  4:     120,      720,      1590,       1530,        541;
  5:     720,     5400,     15960,      23220,      16622,       4683;
  6:    5040,    45360,    168000,     327600,     354606,     201978,      47293;
  7:   40320,   423360,   1884960,    4611600,    6693624,    5762988,    2724878,     545835;
  8:  362880,  4354560,  22649760,   66679200,  121491720,  140268240,  100203270,   40492890,   7087261;
  9: 3628800, 48988800, 291513600, 1003363200, 2200973040, 3190509000, 3055965720, 1864917540, 657944942, 102247563;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+1+(5*n+2)*x+(9*n-1)*x^2+(7*n-4)*x^3+(2*n-2)*x^4)*P(n-1,x) - (x+5*x^2+9*x^3+7*x^4+2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (5*n + 2)*x + (9*n - 1)*x^2 + (7*n - 4)*x^3 + (2*n - 2)*x^4)*P[n - 1, x] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (5*n + 2)*x + (9*n - 1)*x^2 + (7*n - 4)*x^3 + (2*n - 2)*x^4)*v[i - 1] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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

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.