Real-rooted atlas

A008459 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 16\,x - 36\,x^{2} - 32\,x^{3} - 10\,x^{4} + 4\,n + 26\,n x + 50\,n x^{2} + 38\,n x^{3} + 10\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 16\,x^{2} - 36\,x^{3} - 32\,x^{4} - 10\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
26364688
3201804923961088
47084035406120360614176
5252378021560579607278033876190208
6924166321197004384808572208431923245562600704
734327207262647229106007774200118888569602936315189636030464
8128703088803147144177297126022170012590424015775956010804795230915046504047104
94862013127401534104010162152041953111211167556401912328880202703688012047585403055435567104278528
1018475655426807307586055660176027073846808771018256191306715602769026832025414565220133394223603038019876100726755328
117054322327925634179860029431630801648976472063023072112167410138896308615907600386401175160312560459640146872971432303548668561435037302784
122704156973496161575229656151315164009598536090042305724489613267958864162979602642016474970425930052356380076003784711386456160975463313630452334399620526579564544
1310400600405623400717142171276030050096538720782600269119838988097409497099202583839117059250175604815336703833296166006926094191016045244833182640175761347427123065412858696294599134674944
1440116600168489720032314664200374796021600293429590053616389294601680672871763163602060936478057604728091775206808078136021315361011017983935960898177205409120535111020236520191288368499760309468591112564240277467168768
Technical details
Node id
n-6c0d8a4a1303a2d4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-fd95eee1f5b7d105

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,       6;
  2:     6,      36,       46;
  3:    20,     180,      492,       396;
  4:    70,     840,     3540,      6120,      3606;
  5:   252,    3780,    21560,     57960,     72780,      33876;
  6:   924,   16632,   119700,    438480,    857220,     843192,     324556;
  7:  3432,   72072,   626472,   2910600,   7774200,   11888856,    9602936,    3151896;
  8: 12870,  308880,  3147144,  17729712,  60221700,  125904240,  157759560,  108047952,   30915046;
  9: 48620, 1312740, 15341040, 101621520, 419531112, 1116755640, 1912328880, 2027036880, 1204758540, 305543556;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n-2+(26*n-16)*x+(50*n-36)*x^2+(38*n-32)*x^3+(10*n-10)*x^4)*P(n-1,x) - (2*x+16*x^2+36*x^3+32*x^4+10*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[((4*n - 2 + (26*n - 16)*x + (50*n - 36)*x^2 + (38*n - 32)*x^3 + (10*n - 10)*x^4)*P[n - 1, x] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = ((4*n - 2 + (26*n - 16)*x + (50*n - 36)*x^2 + (38*n - 32)*x^3 + (10*n - 10)*x^4)*v[i - 1] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023