Real-rooted atlas

A172431 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(3 - 9\,n + 6\,n^{2}\right)\,P_{n}(x) = \left(-3 - 9\,x - 24\,n - 63\,n x + 24\,n^{2} + 54\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,n - 27\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
2305484
3114396324834
43901998324016207248
5126085322073621060729058878
639243304810724416556411907030618459468
711910119988487620101412011226606123601224723491130
8354664159622032128532494080044206797196293932847239226021832
91040701392804795096025156980479074505563728037843848133844401771470191149302
103018304539186296547481100093402539763103755297703516171121974598565845851064953901388042052
11867096144761041065208324534308721231149780221502859226526210482062935864978205734246824820233834049985444146
122471400453535203711906001783506600557031330011820078900172826362801726330320011399409450464420385010132808408290479671278652736
13699725414000148012615361206753877200238575456005836586868010091815272012346673508010512788715059952449700212788976404062335004290166786505482450414
141969887042677609441985216002478452580097712244000270868260600541106402760784796639400820872625950605857502250302464330740945943722361595917323010044234903564665497020
Technical details
Node id
n-3d19ef0ffa1f54a7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-44bed620cb8d4ddb

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:      6;
  2:     30,      54;
  3:    114,     396,     324;
  4:    390,    1998,    3240,     1620;
  5:   1260,    8532,   20736,    21060,     7290;
  6:   3924,   33048,  107244,   165564,   119070,    30618;
  7:  11910,  119988,  487620,  1014120,  1122660,   612360,   122472;
  8:  35466,  415962, 2032128,  5324940,  8004420,  6797196,  2939328,   472392;
  9: 104070, 1392804, 7950960, 25156980, 47907450, 55637280, 37843848, 13384440, 1771470;
%F Formula
The row polynomials satisfy (6*n^2-9*n+3)*P(n,x) = (24*n^2-24*n-3+(54*n^2-63*n-9)*x)*P(n-1,x) - (5*n+27*n*x+36*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((24*n^2 - 24*n - 3 + (54*n^2 - 63*n - 9)*x)*P[n - 1, x] - (5*n + 27*n*x + 36*n*x^2)*D[P[n - 1, x], x])/(6*n^2 - 9*n + 3)]];
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] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((24*n^2 - 24*n - 3 + (54*n^2 - 63*n - 9)*x)*v[i - 1] - (5*n + 27*n*x + 36*n*x^2)*deriv(v[i - 1], x))/(6*n^2 - 9*n + 3)); 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