Real-rooted atlas

A110441 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 4\,x - 18\,n - 6\,n x + 12\,n^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(-8\,x + 6\,x^{2} + 2\,x^{3} - 6\,n - 66\,n x - 33\,n x^{2} - 3\,n x^{3} + 12\,n^{2} + 56\,n^{2} x + 15\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 6\,x^{3} + x^{4} - 6\,n x - 37\,n x^{2} - 12\,n x^{3} - n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
261420
394645100
41296216124448
5151645948043151892
6182501260279626107627696
72135422957180111157746178530496
8244763780153603384039372215884088118528
92761657962906483790139656128268574489207453872
1030774842450344180180395220522312392232147582204701717568
11339501174581576349650956676167097018106001141533368790450456437568
1236114415840125460627480206438445148326485720590922031933209012969829223937024
133913562079018502010588054078584107611561941400823501745183642208648838216289221297988410432
1442158626676263604169983075139022327925650804376774144548048907054806004228042125112042458738324673792
Technical details
Node id
n-d20569fbd513a05a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-db59e965f0bc5830

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:  3;
  2:  6,  14;
  3:  9,  46,   45;
  4: 12,  96,  216,   124;
  5: 15, 164,  594,   804,   315;
  6: 18, 250, 1260,  2796,  2610,    762;
  7: 21, 354, 2295,  7180, 11115,   7746,   1785;
  8: 24, 476, 3780, 15360, 33840,  39372,  21588,  4088;
  9: 27, 616, 5796, 29064, 83790, 139656, 128268, 57448, 9207;
%F Formula
The row polynomials satisfy (12*n^2-18*n+6+(2*n^2-6*n+4)*x)*P(n,x) = (12*n^2-6*n+(56*n^2-66*n-8)*x+(15*n^2-33*n+6)*x^2+(n^2-3*n+2)*x^3)*P(n-1,x) - (6*n*x+(37*n-1)*x^2+(12*n-6)*x^3+(n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 6*n + (56*n^2 - 66*n - 8)*x + (15*n^2 - 33*n + 6)*x^2 + (n^2 - 3*n + 2)*x^3)*P[n - 1, x] - (6*n*x + (37*n - 1)*x^2 + (12*n - 6)*x^3 + (n - 1)*x^4)*D[P[n - 1, x], x])/(12*n^2 - 18*n + 6 + (2*n^2 - 6*n + 4)*x)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 6*n + (56*n^2 - 66*n - 8)*x + (15*n^2 - 33*n + 6)*x^2 + (n^2 - 3*n + 2)*x^3)*v[i - 1] - (6*n*x + (37*n - 1)*x^2 + (12*n - 6)*x^3 + (n - 1)*x^4)*deriv(v[i - 1], x))/(12*n^2 - 18*n + 6 + (2*n^2 - 6*n + 4)*x)); 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