Real-rooted atlas

A078812 → D → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 2\,n\right)\,P_{n}(x) = \left(4 + 2\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(4\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
10
20
322
412618
5424812102
611221612020468
7252720660240301902
8504198026401560420427146
992447528580728031506725625414
10158410296240242730016800571210087286796
11257420592600608736071400342729576144090287364
1240043861013728024752025704016279263840151201980110928296
136006686402917206364808139606511683351601108802277026401322939556
14873611668858344015116402325600227908814747046375601821603300034321569156204
1512376190944110853633592006104700716284856530323060288113850028600046332436818228127306
Technical details
Node id
n-ed9a32324646062c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5a933c15f37f9bbb

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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:    0;
   2:    0;
   3:    2;
   4:   12,     6;
   5:   42,    48,    12;
   6:  112,   216,   120,    20;
   7:  252,   720,   660,   240,    30;
   8:  504,  1980,  2640,  1560,   420,   42;
   9:  924,  4752,  8580,  7280,  3150,  672,   56;
  10: 1584, 10296, 24024, 27300, 16800, 5712, 1008, 72;
%F Formula
The row polynomials satisfy (2*n-6)*P(n,x) = (2*n+4+(n+2)*x)*P(n-1,x) + (4*x+x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (n + 2)*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((2*n + 4 + (n + 2)*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 6)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023