Real-rooted atlas

A091044 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n x + 2\,n x^{2} - n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(4\,x^{2} - 4\,x^{3} - 4\,n - 22\,n x - 12\,n x^{2} + 12\,n x^{3} + 4\,n^{2} + 14\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x^{3} - 8\,x^{4} - 4\,n x - 16\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2485612116
3256512256321056
4128037123264960809296
56144235523072016128320019279936
628672137216240640190464680969856448675392
7131072753664167116818022409830402580482867210245628928

All 15 rows

Technical details
Node id
n-d48ebb003c7449a7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9c5ecf6080e4ad4f

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:       8,        4;
  2:      48,       56,       12;
  3:     256,      512,      256,        32;
  4:    1280,     3712,     3264,       960,        80;
  5:    6144,    23552,    30720,     16128,      3200,      192;
  6:   28672,   137216,   240640,    190464,     68096,     9856,      448;
  7:  131072,   753664,  1671168,   1802240,    983040,   258048,    28672,    1024;
  8:  589824,  3964928, 10665984,  14696448,  11038720,  4460544,   903168,   79872,   2304;
  9: 2621440, 20185088, 63963136, 107806720, 104366080, 58572800, 18382848, 2973696, 215040, 5120;
%F Formula
The row polynomials satisfy (2*n^3-n^2+(4*n^2-4*n)*x+2*n*x^2)*P(n,x) = (8*n^3+4*n^2-4*n+(12*n^3+14*n^2-22*n)*x+(24*n^2-12*n+4)*x^2+(12*n-4)*x^3)*P(n-1,x) - ((4*n^2+4*n)*x+(8*n^2+16*n)*x^2+(16*n+4)*x^3+8*x^4)*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[((8*n^3 + 4*n^2 - 4*n + (12*n^3 + 14*n^2 - 22*n)*x + (24*n^2 - 12*n + 4)*x^2 + (12*n - 4)*x^3)*P[n - 1, x] - ((4*n^2 + 4*n)*x + (8*n^2 + 16*n)*x^2 + (16*n + 4)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - n^2 + (4*n^2 - 4*n)*x + 2*n*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; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 + 4*n^2 - 4*n + (12*n^3 + 14*n^2 - 22*n)*x + (24*n^2 - 12*n + 4)*x^2 + (12*n - 4)*x^3)*v[i - 1] - ((4*n^2 + 4*n)*x + (8*n^2 + 16*n)*x^2 + (16*n + 4)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - n^2 + (4*n^2 - 4*n)*x + 2*n*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

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