Real-rooted atlas

A086645 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(-26 + 60\,x - 16\,x^{2} + 33\,n - 46\,n x + 8\,n x^{2} - 14\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-10 + 4\,x + 160\,x^{2} - 64\,x^{3} + 17\,n + 62\,n x - 144\,n x^{2} + 32\,n x^{3} - 10\,n^{2} - 48\,n^{2} x + 32\,n^{2} x^{2} + 2\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x - 52\,x^{2} + 128\,x^{3} - 64\,x^{4} - 12\,n x + 64\,n x^{2} - 64\,n x^{3} + 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3549
4131225
525561697
64120080321
761540432641097
885120419044483641
911323526496281625612033
1014541761814414592230439361
1118169004368059520166401024127945
1222110780939841985289856011264413337
13265161041853285660164646409216040961328609
143132319234091214277121793792612352532484251521
15365323965925923267264589388832614404874241638413551753
Technical details
Node id
n-5644542859859603 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-18a4c119033fa265

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   1;
   3:   5,    4;
   4:  13,   12;
   5:  25,   56,    16;
   6:  41,  200,    80;
   7:  61,  540,   432,    64;
   8:  85, 1204,  1904,   448;
   9: 113, 2352,  6496,  2816,  256;
  10: 145, 4176, 18144, 14592, 2304;
%F Formula
The row polynomials satisfy (2*n^3-14*n^2+33*n-26+(8*n^2-46*n+60)*x+(8*n-16)*x^2)*P(n,x) = (2*n^3-10*n^2+17*n-10+(8*n^3-48*n^2+62*n+4)*x+(32*n^2-144*n+160)*x^2+(32*n-64)*x^3)*P(n-1,x) + ((4*n^2-12*n+6)*x-(16*n^2-64*n+52)*x^2-(64*n-128)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 10*n^2 + 17*n - 10 + (8*n^3 - 48*n^2 + 62*n + 4)*x + (32*n^2 - 144*n + 160)*x^2 + (32*n - 64)*x^3)*P[n - 1, x] + ((4*n^2 - 12*n + 6)*x - (16*n^2 - 64*n + 52)*x^2 - (64*n - 128)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 14*n^2 + 33*n - 26 + (8*n^2 - 46*n + 60)*x + (8*n - 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n^3 - 10*n^2 + 17*n - 10 + (8*n^3 - 48*n^2 + 62*n + 4)*x + (32*n^2 - 144*n + 160)*x^2 + (32*n - 64)*x^3)*v[i - 1] + ((4*n^2 - 12*n + 6)*x - (16*n^2 - 64*n + 52)*x^2 - (64*n - 128)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 14*n^2 + 33*n - 26 + (8*n^2 - 46*n + 60)*x + (8*n - 16)*x^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. [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