Real-rooted atlas

A086645 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(240 - 170\,x + 20\,x^{2} - 346\,n + 193\,n x - 18\,n x^{2} + 185\,n^{2} - 70\,n^{2} x + 4\,n^{2} x^{2} - 44\,n^{3} + 8\,n^{3} x + 4\,n^{4}\right)\,P_{n}(x) = \left(39 + 236\,x - 249\,x^{2} + 34\,x^{3} - 92\,n - 489\,n x + 403\,n x^{2} - 46\,n x^{3} + 77\,n^{2} + 349\,n^{2} x - 178\,n^{2} x^{2} + 12\,n^{2} x^{3} - 28\,n^{3} - 108\,n^{3} x + 24\,n^{3} x^{2} + 4\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(-54\,x + 106\,x^{2} - 60\,x^{3} + 8\,x^{4} + 94\,n x - 162\,n x^{2} + 76\,n x^{3} - 8\,n x^{4} - 48\,n^{2} x + 64\,n^{2} x^{2} - 16\,n^{2} x^{3} + 8\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
38210
44536384
5168308964576
64651680117020053520
71056653487123300360619968
8209320020450453203277355887107520
937605168817846421164094640159608968557056
106273117504580788105019277571023990429988129692801664
11988024225016279204199000470288023837405426405244018001013762560
12148614623084061079142120002263261017310216641671811235848662524201166322432
1321528828828922944042167004909309609859768854915168155746802167704136620316812314572800
14302251411280194379901124838003162896754635696003650610601549776803478117539468002073504056131472200704
15413282301390384274802748946209782682001867427730198443448011882379603978374407252245068468403046685096146811549696
Technical details
Node id
n-3d14fead5538beb4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a95c31fc8093635f

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 -2)
  • 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:    0;
   2:    1;
   3:    8,      2;
   4:   45,     36,      3;
   5:  168,    308,     96,       4;
   6:  465,   1680,   1170,     200,      5;
   7: 1056,   6534,   8712,    3300,    360,      6;
   8: 2093,  20020,  45045,   32032,   7735,    588,     7;
   9: 3760,  51688, 178464,  211640,  94640,  15960,   896,    8;
  10: 6273, 117504, 580788, 1050192, 775710, 239904, 29988, 1296, 9;
%F Formula
The row polynomials satisfy (4*n^4-44*n^3+185*n^2-346*n+240+(8*n^3-70*n^2+193*n-170)*x+(4*n^2-18*n+20)*x^2)*P(n,x) = (4*n^4-28*n^3+77*n^2-92*n+39+(12*n^4-108*n^3+349*n^2-489*n+236)*x+(24*n^3-178*n^2+403*n-249)*x^2+(12*n^2-46*n+34)*x^3)*P(n-1,x) + ((8*n^3-48*n^2+94*n-54)*x-(8*n^3-64*n^2+162*n-106)*x^2-(16*n^2-76*n+60)*x^3-(8*n-8)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 28*n^3 + 77*n^2 - 92*n + 39 + (12*n^4 - 108*n^3 + 349*n^2 - 489*n + 236)*x + (24*n^3 - 178*n^2 + 403*n - 249)*x^2 + (12*n^2 - 46*n + 34)*x^3)*P[n - 1, x] + ((8*n^3 - 48*n^2 + 94*n - 54)*x - (8*n^3 - 64*n^2 + 162*n - 106)*x^2 - (16*n^2 - 76*n + 60)*x^3 - (8*n - 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 44*n^3 + 185*n^2 - 346*n + 240 + (8*n^3 - 70*n^2 + 193*n - 170)*x + (4*n^2 - 18*n + 20)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^4 - 28*n^3 + 77*n^2 - 92*n + 39 + (12*n^4 - 108*n^3 + 349*n^2 - 489*n + 236)*x + (24*n^3 - 178*n^2 + 403*n - 249)*x^2 + (12*n^2 - 46*n + 34)*x^3)*v[i - 1] + ((8*n^3 - 48*n^2 + 94*n - 54)*x - (8*n^3 - 64*n^2 + 162*n - 106)*x^2 - (16*n^2 - 76*n + 60)*x^3 - (8*n - 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 44*n^3 + 185*n^2 - 346*n + 240 + (8*n^3 - 70*n^2 + 193*n - 170)*x + (4*n^2 - 18*n + 20)*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