Real-rooted atlas

A103327 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 6\,x - 2\,x^{2} + 5\,n - 16\,n x + 4\,n x^{2} - 8\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(4 + 26\,x + 32\,x^{2} - 20\,x^{3} - 4\,n - 54\,n x - 72\,n x^{2} + 24\,n x^{3} - 16\,n^{2} - 8\,n^{2} x + 48\,n^{2} x^{2} + 16\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 12\,x^{2} + 8\,x^{3} - 16\,x^{4} - 16\,n x^{2} - 32\,n x^{3} - 8\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
24032476
32243521208704
4115228161920384166288
556321894420608832011203254656
62662411468817510411468831360307264465600
71228806471681284096119808053760010752080641283905536

All 15 rows

Technical details
Node id
n-8cfbf83157798e5d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-36aebafdc60acded

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:       6,        2;
  2:      40,       32,        4;
  3:     224,      352,      120,        8;
  4:    1152,     2816,     1920,      384,       16;
  5:    5632,    18944,    20608,     8320,     1120,       32;
  6:   26624,   114688,   175104,   114688,    31360,     3072,      64;
  7:  122880,   647168,  1284096,  1198080,   537600,   107520,    8064,     128;
  8:  557056,  3473408,  8519680, 10452992,  6758400,  2236416,  344064,   20480,   256;
  9: 2490368, 17956864, 52592640, 80510976, 69115904, 33251328, 8515584, 1044480, 50688, 512;
%F Formula
The row polynomials satisfy (4*n^3-8*n^2+5*n-1+(8*n^2-16*n+6)*x+(4*n-2)*x^2)*P(n,x) = (16*n^3-16*n^2-4*n+4+(24*n^3-8*n^2-54*n+26)*x+(48*n^2-72*n+32)*x^2+(24*n-20)*x^3)*P(n-1,x) - ((8*n^2-2)*x+(16*n^2+16*n-12)*x^2+(32*n-8)*x^3+16*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[((16*n^3 - 16*n^2 - 4*n + 4 + (24*n^3 - 8*n^2 - 54*n + 26)*x + (48*n^2 - 72*n + 32)*x^2 + (24*n - 20)*x^3)*P[n - 1, x] - ((8*n^2 - 2)*x + (16*n^2 + 16*n - 12)*x^2 + (32*n - 8)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(4*n^3 - 8*n^2 + 5*n - 1 + (8*n^2 - 16*n + 6)*x + (4*n - 2)*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] = ((16*n^3 - 16*n^2 - 4*n + 4 + (24*n^3 - 8*n^2 - 54*n + 26)*x + (48*n^2 - 72*n + 32)*x^2 + (24*n - 20)*x^3)*v[i - 1] - ((8*n^2 - 2)*x + (16*n^2 + 16*n - 12)*x^2 + (32*n - 8)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(4*n^3 - 8*n^2 + 5*n - 1 + (8*n^2 - 16*n + 6)*x + (4*n - 2)*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