Real-rooted atlas

A091042 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-12 - 53\,x - 86\,x^{2} - 61\,x^{3} - 16\,x^{4} + 8\,n + 34\,n x + 52\,n x^{2} + 34\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 20\,x^{2} - 36\,x^{3} - 28\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
24711
3165241109
4643044722391079
5256160036963736139310681
6102479362432036816275168119105731
7409637888144640291584327264193836473211046629
816384176128804864202624030342402702112132478427580710360559
96553680281642721281289625624149760287223042118454488580961607521102558961
10262144360448021889024770416641731717122577615362540377601598375365825387693693191015229051
11104857615990784109117440438697984115066470420571801602538570496213485440011708561763781424365460839310049731549
1241943047025459253215232024058920967212957696150556672002232419123223513079040172385133448377869264242903431231828103999482086439
13167772163061841922549088256128004915204317806592010306171699217848010342422594028339220749395712013480770272058812496656154691875921855077841984771132841
1467108864132540006412029263872664262410242489830604806689093713921325077757952195958608691221632933376001760681474560102685768960040637386104097806402604108121860079748229241971
1526843545657042534405605267865633757436313613919405998084156735356928927058919424015685749178368202322097684481979654564044814464025648640765183387904027706562800006146179720606301803820196497521286869
Technical details
Node id
n-246f9da6e7934434 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-87f312a3034fd873

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       7;
   3:     16,      52,       41;
   4:     64,     304,      472,      239;
   5:    256,    1600,     3696,     3736,      1393;
   6:   1024,    7936,    24320,    36816,     27516,      8119;
   7:   4096,   37888,   144640,   291584,    327264,    193836,     47321;
   8:  16384,  176128,   804864,  2026240,   3034240,   2702112,   1324784,    275807;
   9:  65536,  802816,  4272128, 12896256,  24149760,  28722304,  21184544,   8858096,  1607521;
  10: 262144, 3604480, 21889024, 77041664, 173171712, 257761536, 254037760, 159837536, 58253876, 9369319;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (8*n-12+(34*n-53)*x+(52*n-86)*x^2+(34*n-61)*x^3+(8*n-16)*x^4)*P(n-1,x) - (4*x+20*x^2+36*x^3+28*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (34*n - 53)*x + (52*n - 86)*x^2 + (34*n - 61)*x^3 + (8*n - 16)*x^4)*P[n - 1, x] - (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (34*n - 53)*x + (52*n - 86)*x^2 + (34*n - 61)*x^3 + (8*n - 16)*x^4)*v[i - 1] - (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

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