Real-rooted atlas

A078812 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-10 - 8\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3810321
4213822485
555130111405341
61444204742566561365
73771308183613245119875461
89873970666660203130924140821845
925841182223109250881643565881554192987381
106765346907737898160776453943012720247225510349525
111771110061025217736610833953520903485855229683762330111398101
1246368289032804228131507213994781013292506408173272393035522418125592405
131213938238002519640458260055046504584620270235011289203289006435078655201322369621
14317811233241877778601557452020845300196358401334382065920802352285593450101530109206371489478485
15832040656629023709783518336007649545080421600619269003534564014951925463385010260251552201483377015357913941
Technical details
Node id
n-91f8c3beb9aeb119 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2cb32fb53e87d6dc

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:    3,     2;
   3:    8,    10,     3;
   4:   21,    38,    22,     4;
   5:   55,   130,   111,    40,     5;
   6:  144,   420,   474,   256,    65,     6;
   7:  377,  1308,  1836,  1324,   511,    98,     7;
   8:  987,  3970,  6666,  6020,  3130,   924,   140,    8;
   9: 2584, 11822, 23109, 25088, 16435,  6588,  1554,  192,   9;
  10: 6765, 34690, 77378, 98160, 77645, 39430, 12720, 2472, 255, 10;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (8*n-10+(6*n-8)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 10 + (6*n - 8)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 10 + (6*n - 8)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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