Real-rooted atlas

A116414 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0) · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(8 + 37\,x + 63\,x^{2} + 47\,x^{3} + 13\,x^{4} - 11\,n x - 35\,n x^{2} - 37\,n x^{3} - 13\,n x^{4}\right)\,P_{n-1}(x) + \left(4 + 28\,x + 73\,x^{2} + 91\,x^{3} + 55\,x^{4} + 13\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
213342269
34016221695513
4121668136512264093789
53642551706096626548176027945
610939294325296007661824336487573206037
7328032804139008323950448880370158168328325851518993
89841113176563422158755627717553072916211402382559814020611198493
9295243838172195880726302415317436213748351974810011654310398832060327582558521
108857312843148303760315487687805399713147457415277174512097806062508555190366702595757608644773
112657204251526306574961315980693738417367383500881035014232103019493071378940032798678089981768111689604487100705
127971611395032411100473153135795017049914103865215024635048269876218625086634422990408579860016903280164219001284805752933080169069
132391484454382033954917402088961694747366928019132922259360713374805072414307653219424024411580278202281107088085818827641964771692206780765243876313161
1471744531470762941390050355803135226831703349467904803680541925922487473106891309203818125193643557997101362475453780021247178211764302194318590969359628897312381797924789621

Computed in your browser ?

Technical details
Node id
n-39b6e7fd4a9cc7c7 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9e17c894e280f681

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

OEIS matches ?

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     4,      5;
  2:    13,     34,      22;
  3:    40,    162,     216,      95;
  4:   121,    668,    1365,    1226,      409;
  5:   364,   2551,    7060,    9662,     6548,     1760;
  6:  1093,   9294,   32529,   60076,    61824,    33648,     7573;
  7:  3280,  32804,  139008,  323950,   448880,   370158,   168328,    32585;
  8:  9841, 113176,  563422, 1587556,  2771755,  3072916,  2114023,   825598,  140206;
  9: 29524, 383817, 2195880, 7263024, 15317436, 21374835, 19748100, 11654310, 3988320, 603275;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = (8-(11*n-37)*x-(35*n-63)*x^2-(37*n-47)*x^3-(13*n-13)*x^4)*P(n-1,x) + (4+28*x+73*x^2+91*x^3+55*x^4+13*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[((8 - (11*n - 37)*x - (35*n - 63)*x^2 - (37*n - 47)*x^3 - (13*n - 13)*x^4)*P[n - 1, x] + (4 + 28*x + 73*x^2 + 91*x^3 + 55*x^4 + 13*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = ((8 - (11*n - 37)*x - (35*n - 63)*x^2 - (37*n - 47)*x^3 - (13*n - 13)*x^4)*v[i - 1] + (4 + 28*x + 73*x^2 + 91*x^3 + 55*x^4 + 13*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-3dedd873ceec398a
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-3dedd873ceec398a
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-3dedd873ceec398a

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023