Real-rooted atlas

A063967 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(12 + 62\,x + 118\,x^{2} + 98\,x^{3} + 30\,x^{4} - 5\,n - 26\,n x - 52\,n x^{2} - 46\,n x^{3} - 15\,n x^{4}\right)\,P_{n-1}(x) + \left(3 + 23\,x + 66\,x^{2} + 90\,x^{3} + 59\,x^{4} + 15\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2268
36324280
416128336288768
5444641816312019807424
61201576820821168270001360871680
732851523348011513622082422377693528692224
8896163841276165486081405056214272018005766428166684672

All 15 rows

Technical details
Node id
n-2a8793360a05669c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2bf4b04ee95e7526

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 -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:    1;
   2:    2,      6;
   3:    6,     32,      42;
   4:   16,    128,     336,     288;
   5:   44,    464,    1816,    3120,     1980;
   6:  120,   1576,    8208,   21168,    27000,    13608;
   7:  328,   5152,   33480,  115136,   220824,   223776,     93528;
   8:  896,  16384,  127616,  548608,  1405056,  2142720,   1800576,    642816;
   9: 2448,  51072,  463680, 2391936,  7665376, 15620736,  19759680,  14179968,   4418064;
  10: 6688, 156768, 1625472, 9782400, 37647040, 96051264, 162417024, 175464576, 109861920, 30365280;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(5*n-12+(26*n-62)*x+(52*n-118)*x^2+(46*n-98)*x^3+(15*n-30)*x^4)*P(n-1,x) + (3+23*x+66*x^2+90*x^3+59*x^4+15*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(5*n - 12 + (26*n - 62)*x + (52*n - 118)*x^2 + (46*n - 98)*x^3 + (15*n - 30)*x^4)*P[n - 1, x] + (3 + 23*x + 66*x^2 + 90*x^3 + 59*x^4 + 15*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(5*n - 12 + (26*n - 62)*x + (52*n - 118)*x^2 + (46*n - 98)*x^3 + (15*n - 30)*x^4)*v[i - 1] + (3 + 23*x + 66*x^2 + 90*x^3 + 59*x^4 + 15*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741