Real-rooted atlas

A110441 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
24161434
38488848192
4161283604161641084
5323201216217618165606120
66476836808960116167552191234552
7128179210368320005664057216304006528195072
8256409627776103936233920323072266432119552222881101328
9512921671680315392862848151782417118721190400461984760966217824
1010242048017971290931229317126279168902720085821445152704176102425980835104288
112048450564403202519040935731223654400414453765023744041210240217520646639744887040198190080
12409698304105881667584002843136082919424171644928253698048265315840191180800899823362481100830285441118931904
1381922129922506752176619528301568027452620865625292811466670081462533120134315008086226739236604723292021632103400966317211264
141638445875258572804515430423456153686759833623537479684752506880716844441680290365446566503424379880243214682675203391447043530329635665403776
Technical details
Node id
n-3bc0a126a17292e8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-89d888d82d5b4636

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 0)
  • 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:   1;
  1:   2,    4;
  2:   4,   16,    14;
  3:   8,   48,    88,     48;
  4:  16,  128,   360,    416,    164;
  5:  32,  320,  1216,   2176,   1816,     560;
  6:  64,  768,  3680,   8960,  11616,    7552,    1912;
  7: 128, 1792, 10368,  32000,  56640,   57216,   30400,    6528;
  8: 256, 4096, 27776, 103936, 233920,  323072,  266432,  119552,  22288;
  9: 512, 9216, 71680, 315392, 862848, 1517824, 1711872, 1190400, 461984, 76096;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(6*n-2)*x+(2*n-2)*x^2)*P(n-1,x) - (x+4*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (6*n - 2)*x + (2*n - 2)*x^2)*P[n - 1, x] - (x + 4*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (6*n - 2)*x + (2*n - 2)*x^2)*v[i - 1] - (x + 4*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n)); 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

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