Real-rooted atlas

A122542 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 18\,x - 18\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 6\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)
\(P_{3}(x) = 12 + 18\,x\)
\(P_{4}(x) = 39 + 126\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
233
3121830
43912681246
51205768103241830
6351223248604212121512870
79967866231663142819440437487270
8276326046966331791721737458310615309576774
97536825123690368689681156680866052336798524883740070
102028325286413232163779784641763066047404010958131220017714723898822
1154012755154452466015197544314247604134304834506486175834804960116590490150939750
121425752208942149086175756637614043213022552344024084118816853896873870299183051901948617944286342
133736086352560476803262080634045851537201111074732145022157012987105847808639762999689206625297863772925861093670
149729511801116014879602872404895623068512455059781964781937793085638895926587565489346735728011847591362359598042072619936138097734
1525202765045423445490912224420667328694277560216418565343868697586650207973768472422705483179611503014867947710457251836482904796066961566221555895270
Technical details
Node id
n-901a4b04aee4ae2d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-daf35d08c219b1f8

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:     3;
   3:    12,     18;
   4:    39,    126,      81;
   5:   120,    576,     810,     324;
   6:   351,   2232,    4860,    4212,    1215;
   7:   996,   7866,   23166,   31428,   19440,    4374;
   8:  2763,  26046,   96633,  179172,  173745,   83106,   15309;
   9:  7536,  82512,  369036,  868968, 1156680,  866052,  336798,   52488;
  10: 20283, 252864, 1323216, 3779784, 6417630, 6604740, 4010958, 1312200, 177147;
%F Formula
The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-4,x) - (4+18*x+18*x^2)*P(n-3,x) - (2+6*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 3, P(3,x) = 12 + 18*x, P(4,x) = 39 + 126*x + 81*x^2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[3, x_] := 12 + 18*x;
P[4, x_] := 39 + 126*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 4, x] - (4 + 18*x + 18*x^2)*P[n - 3, x] - (2 + 6*x + 9*x^2)*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; v[3] = 12 + 18*x; v[4] = 39 + 126*x + 81*x^2; for(n = 5, N, my(i = n); v[i] = -(1 + 6*x + 9*x^2)*v[i - 4] - (4 + 18*x + 18*x^2)*v[i - 3] - (2 + 6*x + 9*x^2)*v[i - 2] + (4 + 6*x)*v[i - 1]); 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

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.