Real-rooted atlas

A046716 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(32 + 16\,x + 2\,x^{2} - 16\,n - 8\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-4 + 4\,n + n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31620541
46417610016356
525617921712576654401
61024240643270416864384432678826
740964311047434245101441735042925219571893481
81638497443841996390416898304756454418900962501001370056341416
9655362634547206191104006186352643409456641120005122194224023749441096011978638881
102621448236695552217433702402494334566416250932224658022553616964304642720396162481747698641079749105326
111048576291528507392852217233408110088037990482520418304039572539699212594746905626612481920360087614428318286898641013622010623401
124194304115093746483203685802285465652852571635712447064981012482471202354380893300371630082437275802112435090063296508069400803505127300108505112182895318578956
1316777216501140928069632174316221418700827441208586403842581507540910080161298628326195270303025439129621815956815872048207833695744743544936012876247051969646791847840130206134510160561511881041
14671088642384952815937126489478767726034944153274379204034560158556293361565696110345629104603136544391818212966401952653630155161651315650912890889807821502699521329732266171521214075488016067031504134816926797486615728464210461906
15268435456123133626314810982449538575177602826249166965910378905600103332648697158696967918048127945801728435623815415719526417706126018902261765384104718311546881225010142271262722059965050501504024887901924654722045996678267201025891787763623697516480540414538467581457001
Technical details
Node id
n-69fddc8e6fad7be5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f00263102480b1ff

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,          2;
   3:     16,         20,           5;
   4:     64,        176,         100,          16;
   5:    256,       1792,        1712,         576,          65;
   6:   1024,      24064,       32704,       16864,        3844,        326;
   7:   4096,     431104,      743424,      510144,      173504,      29252,       1957;
   8:  16384,    9744384,    19963904,    16898304,     7564544,    1890096,     250100,     13700;
   9:  65536,  263454720,   619110400,   618635264,   340945664,  112000512,   21942240,   2374944,   109601;
  10: 262144, 8236695552, 21743370240, 24943345664, 16250932224, 6580225536, 1696430464, 272039616, 24817476, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n-32+(8*n-16)*x+(n-2)*x^2)*P(n-2,x) + (4*n-4+n*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n - 32 + (8*n - 16)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 4 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -(16*n - 32 + (8*n - 16)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 4 + n*x)*v[i - 1]); 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

Same fingerprint ?

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