Real-rooted atlas

A232535 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12 - 15\,x + 6\,x^{2} + 11\,n + 24\,n x - 9\,n x^{2} + 2\,n x^{3} + 7\,n^{2} - 11\,n^{2} x + 4\,n^{2} x^{2} - 6\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-84 - 104\,x + 2\,x^{2} + 17\,x^{3} - 2\,x^{4} + 104\,n + 143\,n x + 12\,n x^{2} - 17\,n x^{3} + 6\,n x^{4} + 4\,n^{2} - 29\,n^{2} x - 15\,n^{2} x^{2} + 12\,n^{2} x^{3} - 24\,n^{3} - 10\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-28\,x - 30\,x^{2} + 8\,x^{3} + 6\,x^{4} - 4\,x^{5} + 16\,n x + 32\,n x^{2} + 8\,n x^{3} - 8\,n x^{4} + 12\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22620349
3128154494335
460810085249652241
52816598444161356165614743
6128003328032128142882966260795729
7573441766402117121250563814457823858615071

All 15 rows

Technical details
Node id
n-c24aa8db59e6190a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d89c48d268675248

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:       5,       2;
  2:      26,      20,       3;
  3:     128,     154,      49,       4;
  4:     608,    1008,     524,      96,       5;
  5:    2816,    5984,    4416,    1356,     165,       6;
  6:   12800,   33280,   32128,   14288,    2966,     260,      7;
  7:   57344,  176640,  211712,  125056,   38144,    5782,    385,     8;
  8:  253952,  905216, 1299456,  964096,  395200,   88992,  10360,   544,   9;
  9: 1114112, 4513792, 7561216, 6779904, 3523840, 1076416, 187776, 17400, 741, 10;
%F Formula
The row polynomials satisfy -(6*n^3-7*n^2-11*n+12-(2*n^3-11*n^2+24*n-15)*x-(4*n^2-9*n+6)*x^2-2*n*x^3)*P(n,x) = -(24*n^3-4*n^2-104*n+84+(10*n^3+29*n^2-143*n+104)*x-(6*n^3-15*n^2+12*n+2)*x^2-(12*n^2-17*n+17)*x^3-(6*n-2)*x^4)*P(n-1,x) + ((12*n^2+16*n-28)*x+(8*n^2+32*n-30)*x^2-(4*n^2-8*n-8)*x^3-(8*n-6)*x^4-4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 4*n^2 - 104*n + 84 + (10*n^3 + 29*n^2 - 143*n + 104)*x - (6*n^3 - 15*n^2 + 12*n + 2)*x^2 - (12*n^2 - 17*n + 17)*x^3 - (6*n - 2)*x^4)*P[n - 1, x] + ((12*n^2 + 16*n - 28)*x + (8*n^2 + 32*n - 30)*x^2 - (4*n^2 - 8*n - 8)*x^3 - (8*n - 6)*x^4 - 4*x^5)*D[P[n - 1, x], x])/(-6*n^3 + 7*n^2 + 11*n - 12 + (2*n^3 - 11*n^2 + 24*n - 15)*x + (4*n^2 - 9*n + 6)*x^2 + 2*n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n^3 - 4*n^2 - 104*n + 84 + (10*n^3 + 29*n^2 - 143*n + 104)*x - (6*n^3 - 15*n^2 + 12*n + 2)*x^2 - (12*n^2 - 17*n + 17)*x^3 - (6*n - 2)*x^4)*v[i - 1] + ((12*n^2 + 16*n - 28)*x + (8*n^2 + 32*n - 30)*x^2 - (4*n^2 - 8*n - 8)*x^3 - (8*n - 6)*x^4 - 4*x^5)*deriv(v[i - 1], x))/(-6*n^3 + 7*n^2 + 11*n - 12 + (2*n^3 - 11*n^2 + 24*n - 15)*x + (4*n^2 - 9*n + 6)*x^2 + 2*n*x^3)); 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