Real-rooted atlas

A084534 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 + 26\,x - 8\,x^{2} + 22\,n - 42\,n x + 8\,n x^{2} - 24\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(6 + 38\,x + 46\,x^{2} - 24\,x^{3} + 2\,n - 22\,n x - 94\,n x^{2} + 24\,n x^{3} - 16\,n^{2} - 40\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 13\,x^{2} + 16\,x^{3} - 16\,x^{4} + 2\,n x - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
258215
372415248
494868242151
51180185150352463
6131203905322854821390
715168707138612744906324105
817224116029924020268878480211967
9192881773569410197100985166118899234524
10213602570990022295297442277092401725120298747
1123440357516082437807407477077471901560924201432280415
122552848122477679254163200216944181896913772516833001682791450
132762463053658213461532752253305257460039782616731039039439419522222093
142972880785216421721761073611802421565904140158281681629229258604573322426210351
Technical details
Node id
n-e801488372546b37 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5ebdabbafcbfe122

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:  3,   2;
  2:  5,   8,    2;
  3:  7,  24,   15,    2;
  4:  9,  48,   68,   24,     2;
  5: 11,  80,  185,  150,    35,     2;
  6: 13, 120,  390,  532,   285,    48,    2;
  7: 15, 168,  707, 1386,  1274,   490,   63,    2;
  8: 17, 224, 1160, 2992,  4020,  2688,  784,   80,  2;
  9: 19, 288, 1773, 5694, 10197, 10098, 5166, 1188, 99, 2;
%F Formula
The row polynomials satisfy (8*n^3-24*n^2+22*n-6+(16*n^2-42*n+26)*x+(8*n-8)*x^2)*P(n,x) = (8*n^3-16*n^2+2*n+6+(24*n^3-40*n^2-22*n+38)*x+(48*n^2-94*n+46)*x^2+(24*n-24)*x^3)*P(n-1,x) - ((4*n^2-2*n-2)*x+(16*n^2-13)*x^2+(32*n-16)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 16*n^2 + 2*n + 6 + (24*n^3 - 40*n^2 - 22*n + 38)*x + (48*n^2 - 94*n + 46)*x^2 + (24*n - 24)*x^3)*P[n - 1, x] - ((4*n^2 - 2*n - 2)*x + (16*n^2 - 13)*x^2 + (32*n - 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 24*n^2 + 22*n - 6 + (16*n^2 - 42*n + 26)*x + (8*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 - 16*n^2 + 2*n + 6 + (24*n^3 - 40*n^2 - 22*n + 38)*x + (48*n^2 - 94*n + 46)*x^2 + (24*n - 24)*x^3)*v[i - 1] - ((4*n^2 - 2*n - 2)*x + (16*n^2 - 13)*x^2 + (32*n - 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 24*n^2 + 22*n - 6 + (16*n^2 - 42*n + 26)*x + (8*n - 8)*x^2)); 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. [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