Real-rooted atlas

A156365 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-4 - 2\,n - 8\,n x - 8\,n x^{2} - 4\,n^{2} x - 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 8\,x^{3} - n x - 4\,n x^{2} + 12\,n x^{3} - 2\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 4\,x + 3\,n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} + n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 8 + 8\,x\)
\(P_{1}(x) = 22 + 88\,x + 24\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
08816
1228824134
252528624641268
311424167248364816013586
42409528579847622419200384163560
549434344374856999616686880948488962191934
6100411686421176169996160141174405609472449792204832410396
7202638272010924608838626562096566401747937284286464020746244608524566250

All 15 rows

Technical details
Node id
n-3e41ebbd7b30e046 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f911b27c2e10a872

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    8,       8;
  1:   22,      88,        24;
  2:   52,     528,       624,         64;
  3:  114,    2416,      7248,       3648,         160;
  4:  240,    9528,     57984,      76224,       19200,         384;
  5:  494,   34344,    374856,     999616,      686880,       94848,         896;
  6: 1004,  116864,   2117616,    9996160,    14117440,     5609472,      449792,        2048;
  7: 2026,  382720,  10924608,   83862656,   209656640,   174793728,    42864640,     2074624,       4608;
  8: 4072, 1221096,  52883712,  623239296,  2515879680,  3739435776,  1974325248,   312600576,    9381888,    10240;
  9: 8166, 3826168, 244504440, 4244382336, 26001965760, 62404717824, 59421352704, 20864378880, 2203872768, 41809920, 22528;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(2*n+4+(4*n^2+8*n)*x+(4*n^3+12*n^2+8*n)*x^2)*P(n-2,x) - ((n+2)*x+(2*n^2+4*n)*x^2-(4*n^2+12*n+8)*x^3)*d/dx P(n-2,x) + (3*n+4+(4*n^2+8*n+4)*x)*P(n-1,x) + ((n+1)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 8 + 8*x, P(1,x) = 22 + 88*x + 24*x^2.
%t Mathematica
P[0, x_] := 8 + 8*x;
P[1, x_] := 22 + 88*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (4*n^2 + 8*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (2*n^2 + 4*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*D[P[n - 2, x], x] + (3*n + 4 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((n + 1)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8 + 8*x; v[2] = 22 + 88*x + 24*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (4*n^2 + 8*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*v[i - 2] - ((n + 2)*x + (2*n^2 + 4*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 2], x) + (3*n + 4 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((n + 1)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.