Real-rooted atlas

A177267 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(n + n x + 2\,n^{2} + 2\,n^{2} x + n^{3} + n^{3} x\right)\,P_{n-2}(x) + \left(6 + 4\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 6 + x\)
\(P_{2}(x) = 24 + 10\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
1617
2241034
3120868214
47207561681644
550407092266118014973
64032071856390546088157318
736288078782456971814797680641876462
8362880093297608501724317046438649625017244
9399168001189569601315716846492367413055736604800369029654
10479001600162706752021252610081316141268377268112381139205962853428
11622702080023786386560359357444642693255074810170199360168171321668428800104802043948
12871782912003703715366406366834970565622347704082655616658726221381830453993456641989642925144
13130767436800061222319424001181872704240012049877055240687127399348521129421690802982827440801089728640040591906601085
142092278988800010710943165440022970496663360026609911293744017844575069871068440893508080136161762724801040210035200885379331627910
153556874280960001977781262284800466965612774720060676656216004804690813309623270216758272421016056432062780656069772509964800232475443200020565604365765270

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 + \frac{1}{3}\,n\right)\,P_{n}(x) = \left(\frac{1}{3}\,n + \frac{1}{3}\,n x + \frac{2}{3}\,n^{2} + \frac{2}{3}\,n^{2} x + \frac{1}{3}\,n^{3} + \frac{1}{3}\,n^{3} x\right)\,P_{n-2}(x) + \left(2 + \frac{4}{3}\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 6 + x\)
\(P_{2}(x) = 24 + 10\,x\)
Technical details
Node id
n-8cad05f13f2c85cb · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ac757e303466262b)
Fingerprint
f6-5e298dbd100ba6ea

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         6,          1;
   2:        24,         10;
   3:       120,         86,          8;
   4:       720,        756,        168;
   5:      5040,       7092,       2661,        180;
   6:     40320,      71856,      39054,       6088;
   7:    362880,     787824,     569718,     147976,      8064;
   8:   3628800,    9329760,    8501724,    3170464,    386496;
   9:  39916800,  118956960,  131571684,   64923674,  13055736,   604800;
  10: 479001600, 1627067520, 2125261008, 1316141268, 377268112, 38113920;
%F Formula
The row polynomials satisfy (n+3)*P(n,x) = (n^3+2*n^2+n+(n^3+2*n^2+n)*x)*P(n-2,x) + (4*n+6)*P(n-1,x) for n >= 3, with P(1,x) = 6 + x, P(2,x) = 24 + 10*x.
%t Mathematica
P[1, x_] := 6 + x;
P[2, x_] := 24 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 2*n^2 + n + (n^3 + 2*n^2 + n)*x)*P[n - 2, x] + (4*n + 6)*P[n - 1, x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 6 + x; v[2] = 24 + 10*x; for(n = 3, N, my(i = n); v[i] = ((n^3 + 2*n^2 + n + (n^3 + 2*n^2 + n)*x)*v[i - 2] + (4*n + 6)*v[i - 1])/(n + 3)); 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