Real-rooted atlas

A207824 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6 + 2\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(8 + 6\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
224832
321214424380
416561704576644000
512102167048544192016039430
684816146928100800342405760384372928
757757612041281037040472320120000161288963428088
83851472938814497338245565120189504038438443008204830863040
92527546670530624855303365887526425163040683827211540481105924608273482250
10163795752514897464715040064575767360296271360101036544228372483299328276480102402393231840
111050721404367403716857494024405302502400319188352013104322563716213767188480090777606758402252820732261492
12668387239225732211160448034860804657098688032111285760154220774405264566272127742976021593088024217600162201649152178107735392
1342219328910177454719360340268147520393599381760305903298240168289118208672006236161962598809641607244806244761606298828838338561064961519412734990
142650896571201207873420640252916955712032224070304002787306958080172848054758479061378457627007992422468906889216129686937601750155264160382976894566422937612884816176000
Technical details
Node id
n-ffe13e658f3623fe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-221150f89a7d7347

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        2;
  2:       24,        8;
  3:      212,      144,       24;
  4:     1656,     1704,      576,       64;
  5:    12102,    16704,     8544,     1920,      160;
  6:    84816,   146928,   100800,    34240,     5760,     384;
  7:   577576,  1204128,  1037040,   472320,   120000,   16128,     896;
  8:  3851472,  9388144,  9733824,  5565120,  1895040,  384384,   43008,   2048;
  9: 25275466, 70530624, 85530336, 58875264, 25163040, 6838272, 1154048, 110592, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (3*n+6+(n+2)*x)*P(n-1,x) + (8+6*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 6 + (n + 2)*x)*P[n - 1, x] + (8 + 6*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + 6 + (n + 2)*x)*v[i - 1] + (8 + 6*x + x^2)*deriv(v[i - 1], x))/(n - 1)); 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. [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