Real-rooted atlas

A008292 → γ → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-3 - 7\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3325
49817
5395416109
6189324136649
71107237015362725285
872811828814976396844513
954351158382162144660487936448861
10448821148057217925849377281768964836601
114085883150901382125137613927632403430435379258743125
12405331291654713362648531522066048647787404811184128766520657
1343584795919501513023502718640318666528015089164803330347522236825610939702669
145045745069245467626124885294356050683161600287177126408338355712951878656167136559849
15625948290273290069626747189685893448388394020005557712140802033364648963575195443219037573122746173173765
Technical details
Node id
n-43ff866bc7d00d30 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0986952e4efc838b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (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:      1;
   2:      1;
   3:      3,       2;
   4:      9,       8;
   5:     39,      54,      16;
   6:    189,     324,     136;
   7:   1107,    2370,    1536,    272;
   8:   7281,   18288,   14976,   3968;
   9:  54351,  158382,  162144,  66048,   7936;
  10: 448821, 1480572, 1792584, 937728, 176896;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-3+(2*n-4)*x)*P(n-1,x) - (3+7*x+4*x^2)*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[(2*n - 3 + (2*n - 4)*x)*P[n - 1, x] - (3 + 7*x + 4*x^2)*D[P[n - 1, x], x]];
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] = (2*n - 3 + (2*n - 4)*x)*v[i - 1] - (3 + 7*x + 4*x^2)*deriv(v[i - 1], x)); 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